Clay Mathematics Proceedings Volume 8, 2009

Rational surfaces over nonclosed fields

Brendan Hassett

Abstract. This paper is based on lectures given at the Clay Summer School on Arithmetic Geometry in July 2006.

These notes offer an introduction to the birational geometry of algebraic surfaces, emphasizing the aspects useful for arithmetic. The first three sections are explicitly devoted to birational questions, with a special focus on rational surfaces. We explain the special rôle these play in the larger classification theory. The geometry of rational ruled surfaces and Del Pezzo surfaces is studied in substantial detail. We extend this theory to geometrically rational surfaces over non-closed fields, enumerating the minimal surfaces and describing their geometric properties. This gives essentially the complete classification of rational surfaces up to birational equivalence.

The final two sections focus on singular Del Pezzo surfaces, universal torsors, and their algebraic realizations through Cox rings. Current techniques for counting rational points (on rational surfaces over number fields) often work better for singular surfaces than for smooth surfaces. The actual enumeration of the rational points often boils down to counting integral points on the universal torsor. Universal torsors were first employed in the (ongoing) search for effective criteria for when rational surfaces over number fields admit rational points.

It might seem that these last two topics are far removed from birational geometry, at least the classical formulation for surfaces. However, singularities and finite-generation questions play a central rôle in the minimal model program. And the challenges arising from working over non-closed fields help highlight structural characteristics of this program that usually are only apparent over ℂ in higher dimensions. Indeed, these notes may be regarded as an arithmetically motivated introduction to modern birational geometry.

In general, the prerequisites for these notes are a good understanding of algebraic geometry at the level of Hartshorne [Har77]. Some general understanding of descent is needed to appreciate the applications to non-closed fields. Readers

\begin{matrix}c \\ ◯ \end{matrix}2009 Brendan Hassett

2000 Mathematics Subject Classification. Primary 14J26; Secondary 14G05, 14J20.

Key words and phrases. rational surfaces, Del Pezzo surfaces, minimal models.

Supported in part by NSF Grants #0134259 and #0554491, the Clay Mathematics Institute, and the University of Göttingen.

interested in applications to positive characteristic would benefit from some exposure to étale cohomology at the level of Milne [Mil80]. There is one place where we do not fully observe these prerequisites: The discussion of the Cone Theorem is not self-contained although we do sketch the main ideas. Thankfully, a number of books ([CKM88], [KM98], [Rei97], [Kol96],[Mat02]) give good introductory accounts of this important topic.

Finally, we should indicate how this account relates to others in the literature. The general approach taken to the geometry of surfaces over algebraically closed fields owes much to Beauville’s book [Bea96] and Reid’s lecture notes [Rei97]. The extensions to non-closed fields draw from Kollár’s book [Kol96]. Readers interested in details of the Galois action on the lines of a Del Pezzo surface and its implications for arithmetic should consult Manin’s classic book [Man74] and the more recent survey [MT86]. The books [CKM88, KM98, Mat02] offer a good introduction to modern birational geometry; [Laz04] has a comprehensive account of linear series. We have made no effort to explain how universal torsors and Cox rings are used for the descent of rational points; the recent book of Skorobogatov [Sko01] does a fine job covering this material.

I am grateful to Anthony Várilly-Alvarado, Michael Joyce, Ambrus Pál, and other members of the summer school for helpful comments.

1. Rational surfaces over algebraically closed fields

Let k be an algebraically closed field. Throughout, a variety will designate an integral separated scheme of finite type over k.

1.1. Classical example: Cubic surfaces. Here a cubic surface means a smooth cubic hypersurface X ⊂ ℙ^{3}. We recall a well-known construction for such surfaces:

Let p_{1}, …, p_{6} ∈ ℙ^{2} be points in the projective plane in general position, i.e., • the points are distinct;

• no three of the points are collinear;

• the six points do not lie on a plane conic. Consider the vector space of homogeneous cubics vanishing at these points; it is an exercise to show this has dimension four

I_{p_{1},...,p_{6}}(3) = ⟨F_{0}, F_{1}, F_{2}, F_{3}⟩

and has no additional basepoints.

The resulting linear series gives a rational map

\begin{matrix}ρ : ℙ^{2} & −−→ & ℙ^{3} \\ [x_{0}, x_{1}, x_{2}] & ↦ & [F_{0}, F_{1}, F_{2}, F_{3}] \end{matrix}

that is not well-defined at p_{1}, …, p_{6}. Consider the blow-up

β : X := Bl_{p_{1},...,p_{6}}ℙ^{2} → ℙ^{2}

with exceptional divisors

E_{1}, …, E_{6}.

Blowing up the base scheme of a linear series resolves its indeterminacy, so we obtain a morphism

j : X → ℙ^{3}

with j = ρ ∘ β.

Proposition 1.1. The morphism j gives a closed embedding of X in ℙ^{3}.

We leave the proof as an exercise.

Given this, we may describe the image of j quite easily. The first step is to analyze the Picard group Pic(X) and its associated intersection form

\begin{matrix}Pic(X) × Pic(X) & → & ℤ \\ (D_{1}, D_{2}) & ↦ & D_{1} · D_{2} \end{matrix} .

We recall what happens to the intersection form under blow-ups. Let β : Y → P be the blow-up of a smooth surface at a point, with exceptional divisor E. Then we have an orthogonal direct-sum decomposition

Pic(Y) = Pic(P) ⊕_{⊥} ℤE, E · E = E^{2} = −1,

where the inclusion Pic(P) ↪ Pic(Y) is induced by β^{∗}.

Returning to our particular situation, we have

Pic(X) = ℤL ⊕ ℤE_{1} ⊕ ⋯ ⊕ ℤE_{6}.

Here L is the pullback of the hyperplane from ℙ^{2} with L^{2} = L·L = 1 and L·E_{a} = 0 for each a. We also have E_{a} · E_{b} = 0 for a ≠ b.

Since j is induced by the linear series of cubics with simple basepoints at p_{1}, …, p_{6}, we have

j^{∗}O_{ℙ^{3}}(1) = O_{X}(3L − E_{1} − ⋯ − E_{6})

so that

deg(j(X)) = (3L − E_{1} − ⋯ − E_{6})^{2} = 9 − 6 = 3.

This proves that the image is a smooth cubic surface. The images of the exceptional divisors E_{1}, …, E_{6} have degree

E_{i} · (3L − E_{1} − ⋯ − E_{6}) = 1

and thus are lines on our cubic surface.

Proposition 1.2. The cubic surface j(X) ⊂ ℙ^{3} contains the following 27 lines: • the exceptional curves E_{a};

• proper transforms of lines through p_{a} and p_{b}, with class L − E_{a} − E_{b};

• proper transforms of conics through five basepoints p_{a}, p_{b}, p_{c}, p_{d}, p_{e}, with class 2L − E_{a} − E_{b} − E_{c} − E_{d} − E_{e}.

This beautiful analysis leaves open a number of classification questions:

(1) Does every cubic surface arise as the blow-up of ℙ^{2} in six points in general linear position?

(2) Are there exactly 27 lines on a cubic surface? To address these, we introduce some general geometric definitions:

Definition 1.3. Let Y be a smooth projective surface with canonical class K_{Y}, i.e., the divisor class associated with the differential two-forms Ω_{Y}^{2} = ⋀^{2} Ω_{Y}^{1} . We say that Y is a Del Pezzo surface if −K_{Y} is ample, i.e., there exists an embedding Y ⊂ ℙ^{N} such that O_{ℙ^{N}} (1)|Y = O_{Y} (−rK_{Y}) for some r > 0.

Note that if Y is Del Pezzo then K_{Y}^{2} > 0.

Remark 1.4. Let X ⊂ ℙ^{3} be a cubic surface and H the hyperplane class on ℙ^{3}. Adjunction

K_{X} = (K_{ℙ^{3}} + X)|_{X} = (−4H + 3H)|_{X} = −H|_{X}

implies that any cubic surface is a Del Pezzo surface.

Definition 1.5. Let Y be a smooth projective surface. A (−1)-curve is a smooth rational curve E ⊂ Y with E^{2} = −1.

Of course, exceptional divisors are the main examples. We also have the following characterization.

Proposition 1.6. Let Y be a smooth projective surface. Let E ⊂ Y be an irreducible curve with

E^{2} < 0, K_{Y} · E < 0.

Then E is a (−1)-curve. In particular, on a Del Pezzo surface every irreducible curve with E^{2} < 0 is a (−1)-curve.

Proof. Let p_{a}(E) denote the arithmetic genus of E. Since E is an irreducible curve we know that p_{a}(E) ≥ 0 with equality if and only if E ≃ ℙ^{1}. Combining this with the adjunction formula, we obtain

−2 ≤ 2p_{a}(E) − 2 = E · (K_{Y} + E).

Thus E^{2} = −1, K_{Y} · E = −1, and E is a smooth rational curve. □

Remark 1.7. The lines on a cubic surface are precisely its (−1)-curves. Indeed, if ℓ ⊂ X is a line then the genus formula gives

−2 = 2g(ℓ) − 2 = ℓ^{2} + K_{X} · ℓ = ℓ^{2} − 1.

Suppose then that

ℓ = aL − b_{1}E_{1} − ⋯ − b_{6}E_{6}

is a line on a cubic surface. Then the following equations must be satisfied

\begin{aligned}1 & = & −K_{X} · ℓ = 3a − b_{1} − b_{2} − b_{3} − b_{4} − b_{5} − b_{6} \\ −1 & = & ℓ^{2} = a^{2} − b_{1}^{2} − b_{2}^{2} − b_{3}^{2} − b_{4}^{2} − b_{5}^{2} − b_{6}^{2} \end{aligned}

and these can be solved explicitly. There are precisely 27 solutions; see Exercise 1.1.6 and [Man74, 26.2], especially for the connection with root systems. Thus the cubic surfaces arising as blow-ups of ℙ^{2} in six points in general position have precisely 27 lines.

We extend this analysis to all smooth cubic surfaces:

Theorem 1.8. Let ℙ^{19} = ℙ(Sym^{3}(k^{4})) parametrize all cubic surfaces and let

Z = {(X, ℓ) : X cubic surface, ℓ ⊂ X line } ⊂ ℙ^{19} × 𝔾(1, 3)

denote the incidence correspondence. Let U ⊂ ℙ^{19} denote the locus of smooth cubic surfaces and

π_{1} : Z_{U} := Z ×_{ℙ^{19}} U → U

the projection. Then π_{1} is a finite étale morphism.

Since U is connected the degree of π_{1} is constant, and we conclude

Corollary 1.9. Every smooth cubic surface has 27 lines.

Proof. (cf. [Mum95, p. 173 ff.]) We claim that Z is proper and irreducible of dimension 19: Each line ℓ is contained in a 20 − 4 = 16-dimensional linear series of cubic surfaces, so the projection Z → 𝔾(1, 3) is a ℙ^{15} bundle. Consequently, π_{1} is a proper morphism. In particular, for each one-parameter family of lines in cubic surfaces (X_{t}, ℓ_{t}), the flat limit

lim_{t→0}(X_{t}, ℓ_{t})

is also a line in a cubic surface.

Let N_{ℓ/X} denote the normal bundle of a line ℓ in a smooth cubic surface X. We have N_{ℓ/X} ≃ O_{ℙ^{1}}(−1) so that

h^{0}(N_{ℓ/X}) = h^{1}(N_{ℓ/X}) = 0.

Recall that H^{0}(N_{ℓ/X}) (resp. H^{1}(N_{ℓ/X})) is the tangent space (resp. obstruction space) of the scheme of lines on X at ℓ. It follows then that Z_{U} is smooth of relative dimension zero over U, i.e., π_{1} is étale. Furthermore, proper étale morphisms are finite. □

One further piece of information can be extracted from this result: The intersections of the 27 lines are constant over all the cubic surfaces. This means that every cubic surface X contains a pair (and even a sextuple!) of pairwise disjoint lines (cf. Exercise 1.1.3).

Proposition 1.10. Let X be a smooth cubic surface containing disjoint lines E_{1} and E_{2}. Let ℓ_{1}, …, ℓ_{5} denote the lines in X meeting E_{1} and E_{2}. There is a birational morphism

\begin{matrix}φ : X & → & ℙ^{1} × ℙ^{1} \\ x & ↦ & (p_{E_{1}}(x), p_{E_{2}}(x)) \end{matrix}

where p_{E_{i}} : ℙ^{3} −−→ ℙ^{1} is projection from E_{i}. This contracts ℓ_{1}, …, ℓ_{5} to distinct points q_{1}, …, q_{5} ∈ ℙ^{1} × ℙ^{1} satisfying the following genericity conditions:

• no pair of them lie on a ruling of ℙ^{1} × ℙ^{1};

• no four of them lie on a curve of bidegree (1, 1); The inverse φ^{−1} is given by the linear system of forms of bidegree (2, 2) through q_{1}, …, q_{5}.

We leave this as an exercise.

Corollary 1.11. Every smooth cubic surface is isomorphic to ℙ^{2} blown up at six points.

Proof. We first verify that Bl_{q_{1},...,q_{5}}ℙ^{1} × ℙ^{1} is isomorphic to ℙ^{2} blown up at six points. Indeed, we can realize ℙ^{1} × ℙ^{1} as a smooth quadric Q ⊂ ℙ^{3}, so that the fibers of each projection are lines on Q. Let q ∈ Q be any point and R_{1} and R_{2} the two rulings passing though q. Projection from q

p_{q} : Q −−→ ℙ^{2}

lifts to a morphism

Bl_{q}Q → ℙ^{2}

contracting the proper transforms of R_{1} and R_{2}. □

Before concluding, we draw two morals from this story:

• (−1)-curves govern much of the geometry of a Del Pezzo surface;

• classifying (−1)-curves is a crucial step in classifying the surfaces.

Exercises.

Exercise 1.1.1. Show that six distinct points on the plane impose independent conditions on cubics if no four of the points are collinear. Show that the resulting linear system has base scheme equal to these six points.

Exercise 1.1.2. Give a careful proof of Proposition 1.1.

Exercise 1.1.3. Verify that the 27 curves described in Proposition 1.2 are in fact lines on the cubic surface. Check that each of these has self-intersection −1. Show that

(1) each line is intersected by ten other lines;

(2) any pair of disjoint lines is intersected by five lines;

(3) each line is contained in a collection of six pairwise disjoint lines.

Exercise 1.1.4. Prove Proposition 1.10.

Exercise 1.1.5. Let X be a smooth cubic surface. Show that the intersection form on K_{X}^{⊥} ⊂ Pic(X) is isomorphic to

\begin{aligned}& ρ_{1} & ρ_{2} & ρ_{3} & ρ_{4} & ρ_{5} & ρ_{6} \\ ρ_{1} & −2 & 1 & 0 & 0 & 0 & 0 \\ ρ_{2} & 1 & −2 & 1 & 0 & 0 & 0 \\ ρ_{3} & 0 & 1 & −2 & 1 & 0 & 1 \\ ρ_{4} & 0 & 0 & 1 & −2 & 1 & 0 \\ ρ_{5} & 0 & 0 & 0 & 1 & −2 & 0 \\ ρ_{6} & 0 & 0 & 1 & 0 & 0 & −2 \end{aligned} .

Up to sign, this is the Cartan matrix associated to the root system E_{6}.

Exercise 1.1.6. Consider a line on a cubic surface ℓ ⊂ X, and the associated class λ = 3ℓ + K_{X} ∈ K_{X}^{⊥}. Verify that λ^{2} = −12 and λ · η ≡ 0 (mod 3) for each η ∈ K_{X}^{⊥} ⊂ Pic(X). Deduce that there are a finite number of lines on a cubic surface.

1.2. The structure of birational morphisms of surfaces. Our first task is to show that all (−1)-curves arise as exceptional curves of blow-ups:

Theorem 1.12 (Castelnuovo contraction criterion). [Har77, V.5.7] Let X be a smooth projective surface and E ⊂ X a (−1)-curve. Then there exists a smooth projective surface Y and a morphism β : X → Y contracting E to a point y ∈ Y, so that X is isomorphic to Bl_{y}Y. Each morphism ψ : X → Z contracting E admits a factorization

ψ : X →^{β} Y → Z.

Proof. (Sketch) Let H be a very ample divisor on X such that

H^{1}(X, O_{X}(H)) = 0.

Set L = O_{X}(H +(H ·E)E) so that L|E ≃ O_{E}. For each n > 0 we have the inclusion

O_{X}(nH) ↪ L^{n} = O_{X}(nH + n(H · E)E)

which is an isomorphism away from E. Thus the sections in the image of

Γ(X, O_{X}(nH)) ↪ Γ(X, L^{n})

induce an embedding of X \ E.

We claim that L is globally generated, so we have a morphism

β : X → Y := Proj(⨁_{n≥0} Γ(X, L^{n})).

Since L|E is trivial, β necessarily contracts E to a point; β is an isomorphism away from E.

Here is the idea: Since L|E is globally generated, it suffices to show that the restriction

Γ(X, L) → Γ(E, L|E) ≃ Γ(ℙ^{1}, O_{ℙ^{1}})

is surjective. Taking the long exact sequence associated to

0 → L(−E) → L → L|E → 0,

we are reduced to showing that H^{1}(X, L(−E)) = 0. Indeed, we can show inductively that H^{1}(X, O_{X}(H + aE)) = 0 for a = 1, …, H · E − 1: The exact sequence

0 → O_{X}(H + (a − 1)E) → O_{X}(H + aE) → O_{E}(H · E − a) → 0

expresses O_{X}(H + aE) as an extension of sheaves with vanishing H^{1}.

The trickiest bit is to check that Y is smooth and β is the blow-up of a point of Y. The necessary local computation can be found in [Bea96, II.17] or [Har77, pp. 415].

For the factorization step, the standard isomorphism

Pic(X) = β^{∗}Pic(Y) ⊕ ℤE

identifies β^{∗}Pic(Y) with line bundles on X restricting to zero along E. Moreover, the induced map

Γ(Y, M′) → Γ(X, β^{∗}M′)

is an isomorphism. Suppose that M is very ample on Z so that ψ is induced by certain sections of ψ^{∗}M. However, M = β^{∗}M′ for some M′ on Y and the relevant sections of ψ^{∗}M come from sections of M′. □

Theorem 1.13. Let ϕ : X → Y be a birational morphism of smooth projective surfaces. Then there exists a factorization

X = X_{0} →^{β_{1}} X_{1} → ⋯ → X_{r−1} →^{β_{r}} X_{r} = Y

where each β_{j} is a blow-up of a point on X_{j}. (If ϕ is an isomorphism we take X_{0} = X_{r}.)

Proof. We assume ϕ is not an isomorphism. Hence it is ramified and the induced map

ϕ^{∗}Ω_{Y}^{2} → Ω_{X}^{2}

is not an isomorphism. Since these sheaves are invertible, we can therefore write

ϕ^{∗}Ω_{Y}^{2} = Ω_{X}^{2}(−(m_{1}E_{1} + ⋯ + m_{r}E_{r})),

where the E_{i} are irreducible ϕ-exceptional curves, i.e., ϕ contracts E_{i} to a point in Y. Since ϕ^{∗}K_{Y} |_{E_{i}} is trivial we have ϕ^{∗}K_{Y} · E_{i} = 0. The multiplicity m_{i} is positive because ϕ is ramified along E_{i}. In divisorial notation, we obtain the discrepancy formula:

(1.1)  K_{X} = ϕ^{∗}K_{Y} + ∑ m_{i}E_{i}, m_{i} > 0.

By the Hodge index theorem [Har77, V.1.9], the intersection form on

Λ = ℤE_{1} + ⋯ + ℤE_{r}

is negative definite, so in particular each E_{i}^{2} < 0. We claim that K_{X} · E_{i} < 0 for some i; then Proposition 1.6 guarantees that E_{i} is a (−1)-curve.

We know that

(∑ m_{i}E_{i})^{2} = ∑_{i} m_{i}E_{j} · (∑_{j} m_{j}E_{j}) = ∑_{i} m_{i}E_{i} · K_{X}

is negative, because the intersection form on Λ is negative definite. Hence some E_{i} · K_{X} must be negative.

Using the Castelnuovo criterion we contract E_{i}

X = X_{0} → X_{1},

so that X_{0} is the blow-up of X_{1} at a point. Moreover, ϕ factors through X_{1}. This factorization process terminates because the exceptional locus of ϕ has a finite number of irreducible components. □

Definition 1.14. A smooth projective surface X is minimal if every birational morphism ϕ : X → Y to a smooth variety is an isomorphism.

Theorem 1.13 says that X is minimal if and only if it has no (−1)-curves.

Exercises.

Exercise 1.2.1. Let X be the blow-up of ℙ^{2} at [0, 0, 1], [0, 1, 0], [1, 0, 0]. Realize X in ℙ^{2} × ℙ^{2} ⊂ ℙ^{8} using the bihomogeneous equations

x_{0}y_{0} = x_{1}y_{1} = x_{2}y_{2}.

Verify that the proper transforms of the lines x_{0} = 0, x_{1} = 0, x_{2} = 0 are (−1)-curves and write down explicit linear series contracting each one individually.

Exercise 1.2.2. Let X be a cubic surface, realized as ℙ^{2} blown up at six points. Describe a basepoint-free linear series on X contracting the six curves

2L − E_{a} − E_{b} − E_{c} − E_{d} − E_{e}.

What is the image of the corresponding morphism X → Y?

1.3. Relative minimality and ruled surfaces. Let f : X → B denote a dominant morphism from a smooth projective surface to a variety. We say that X is minimal relative to f if there exists no commutative diagram

X  ⟶^{ϕ}  Y
↘  ↙
B

where ϕ is birational and Y is smooth. In analogy to Theorem 1.13, X is minimal relative to f if and only if there are no (−1)-curves in the fibers of f.

A ruled surface is a morphism f : X → B from a smooth projective surface to a smooth curve whose generic fiber is rational; it is minimal if it is minimal relative to f. If f is smooth then each fiber is isomorphic to ℙ^{1}; in this case, f : X → B is called a ℙ^{1}-bundle.

Proposition 1.15. Let X be a smooth projective surface and f : X → B a ℙ^{1}-bundle. Then each b ∈ B admits an étale-open neighborhood U → B and an isomorphism:

X ×_{B} U →^{∼} ℙ^{1} × U
↘  ↙
U

Proof. Since f is smooth it admits a multisection M ⊂ X with f|M unbranched over b; let U ⊂ M denote the open set where f is unramified. The pullback g : X′ := X×_{B} M → M admits the canonical diagonal section Σ. Consider the direct images of O_{X′}(Σ). Cohomology and base change implies that ℝ^{1}g_{∗}O_{X′}(Σ) is trivial and E := g_{∗}O_{X′}(Σ) has rank two. Under these conditions cohomology commutes with base change, so a fiber-by-fiber analysis shows that

g^{∗}g_{∗}O_{X′}(Σ) → O_{X′}(Σ)

is surjective and the induced morphism

X′ → ℙ(E)

is an isomorphism over M. □

Theorem 1.16. Let f : X → B be a minimal ruled surface. Then X is a ℙ^{1}-bundle over B.

Before proving this, we’ll require a preliminary result.

Lemma 1.17. Let F denote the class of a fiber of f. Consider a fiber of f with irreducible components E_{1}, …, E_{r}. Then we have E_{i}^{2} < 0 and F · E_{i} = 0 for each i and K_{X} · E_{i} < 0 for some i. In particular, each reducible fiber contains a (−1)-curve.

Proof. Each fiber of f is numerically equivalent to F, i.e., has the same intersection numbers with curves in X. Since these fibers are generally disjoint from the E_{i}, we have F · E_{i} = 0 for each i and F · F = 0.

Express F = ∑_{i=1}^{r} m_{i}E_{i} where m_{i} > 0 is the multiplicity of the fiber along E_{i}. Note that F is connected, e.g., by Stein factorization. Thus each E_{i} meets some E_{j} and

E_{i} · ∑_{j≠i} m_{j}E_{j} = E_{i} · (F − m_{i}E_{i}) > 0.

It follows that E_{i} · E_{i} < 0.

Finally, K_{X} · F = −2 by adjunction, so K_{X} · E_{i} < 0 for some index. □

Proof. (Theorem 1.16)

The key point is to show that the fibers of f are all isomorphic to ℙ^{1}. Since f is a dominant morphism from a nonsingular surface to a nonsingular curve, it is flat with fibers of arithmetic genus zero. Each fiber is a Cartier divisor on X and thus has no embedded points. Under the assumptions of Theorem 1.16, each fiber of f is irreducible. We also have that each fiber has multiplicity one. Indeed, writing F = mE we have

−2 = K_{X} · F = mK_{X} · E

so m = 1, 2. However, if m = 2 then adjunction yields 2g(E)−2 = E^{2}+K_{X}E = −1, which is absurd. Thus each fiber of f is isomorphic to ℙ^{1}, and in particular f is smooth. □

We record one additional fact for future reference, whose proof is left as an exercise:

Proposition 1.18. Let E_{1}, …, E_{r} be the components of a fiber of a ruled surface; let F denote the class of the fiber. The induced intersection form on

(ℤE_{1} + ⋯ + ℤE_{r})/ℤF

is negative definite and unimodular.

We now pursue a finer analysis of the structure of ruled surfaces.

Proposition 1.19. Let C be a variety defined over a field K such that CK̄ ≃ ℙ1 ; K̄. Then there exists a closed embedding C ↪ ℙ^{2} as a plane conic. There is a quadratic extension K′/K such that C_{K′} ≃ ℙ_{K′}^{1}.

We leave the proof as an exercise.

We apply Proposition 1.19 in the case where K is the function field of the base B and C is the generic fiber f:

C  ⊂ X
f_{∘} ↓  ↓ f
Spec(k(B)) → B

The Tsen-Lang theorem says that every quadratic form in ≥ 3 variables over k(B) represents zero, so C(k(B)) ≠ ∅. Each rational point corresponds to a section Spec(k(B)) → C of f_{∘}, and thus to a rational map from B to X. Since X is proper, this extends uniquely to a section s : B → X of f. We have proven the following:

Proposition 1.20. Let f : X → B be a ruled surface. There exists a section s : B → X of f.

Combining this with the argument for Proposition 1.15, we obtain

Corollary 1.21 (Classification of ruled surfaces). Every minimal ruled surface f : X → B is isomorphic to ℙ(E) for some rank-two vector bundle E on B.

Combining this with Grothendieck’s classification of vector bundles on ℙ^{1} gives:

Corollary 1.22. Every ruled surface f : X → ℙ^{1}, minimal relative to f, is isomorphic to a Hirzebruch surface

𝔽_{d} := ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−d)), d ≥ 0.

In particular, ruled surfaces over ℙ^{1} are rational.

See Exercise 1.3.3 for more details of the argument.

Exercises.

Exercise 1.3.1. Prove Proposition 1.19. Hint: Note that Ω_{C}^{1} is an invertible sheaf on C defined over K and coincides with O_{ℙ^{1}}(−2) over CK̄. Use the sections of the dual (Ω_{C}^{1})^{∗} to embed C.

Exercise 1.3.2. Prove Proposition 1.18. Hint: Use the mechanism of the proof of Theorem 1.16.

Exercise 1.3.3. Give a detailed proof for Corollary 1.22. For the classification assertion, show that each vector bundle E on ℙ^{1} decomposes as

E ≃ ⨁_{i=1}^{r} O_{ℙ^{1}}(a_{i}), a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{r}.

To establish rationality, exhibit a nonempty open subset U ⊂ ℙ^{1} such that E|U ≃ O_{U}^{⊕r}.

Exercise 1.3.4. For each d ≥ 0, show there exists a diagram

Y
↙^{β}  ↘^{γ}
𝔽_{d}  𝔽_{d+1}

where β (resp. γ) is the blow-up of 𝔽_{d} (resp. 𝔽_{d+1}) at a suitable point.

2. Effective cones and classification

From the modern point of view, the presence of (−1)-curves is controlled by how the effective cone and the canonical class interact. In this section, we develop technical tools for analyzing this interaction. We continue to work over an algebraically closed field k.

2.1. Cones of curves and divisors. Let X be a smooth projective complex variety, N_{1}(X, ℤ) ⊂ H_{2}(X, ℤ) the sublattice generated by homology classes of algebraic curves in X, and N^{1}(X, ℤ) ⊂ H^{2}(X, ℤ) the Néron-Severi group parametrizing homology classes of divisors in X.

We can extend these definitions to fields of positive characteristic: Consider the Chow group of dimension (resp. codimension) one cycles in X; two cycles are numerically equivalent if their intersections with any divisor (resp. curve) are equal. We define N_{1}(X, ℤ) (resp. N^{1}(X, ℤ)) as the quotient of the corresponding Chow group by the cycles numerically equivalent to zero. The rank of N^{1}(X, ℤ) is bounded by the second (étale) Betti number of X; see [Mil80, V.3.28] for the surface case.

Definition 2.1. A Cartier divisor D on a variety X is nef (numerically eventually free or numerically effective) if D · C ≥ 0 for each curve C ⊂ X.

Here is the main example: A Cartier divisor D is semiample if ND is basepointfree for some N ∈ ℕ. Since ND remains basepoint-free when restricted to curves C ⊂ X, we have D · C = deg D|C ≥ 0.

We have the monoid of effective curves

NE_{1}(X, ℤ) = {[D] ∈ N_{1}(X, ℤ) : D effective sum of curves }

and the associated closed cone

\overline{NE}_{1}(X) = smallest closed cone containing NE_{1}(X, ℤ) ⊂ N_{1}(X, ℝ),

as well as the monoid of effective divisors

NE^{1}(X, ℤ) = {[D] ∈ N^{1}(X, ℤ) : D effective divisor }

and the associated cone of pseudo-effective divisors

\overline{NE}^{1}(X) = smallest closed cone containing NE^{1}(X, ℤ) ⊂ N^{1}(X, ℝ).

We also have the nef cone \overline{NM}^{1}(X) ⊂ N^{1}(X, ℝ) and NM_{∘}^{1}(X) its interior. Note that \overline{NM}^{1}(X) and \overline{NE}_{1}(X) are dual in the sense that

\overline{NM}^{1}(X) = \{D ∈ N^{1}(X, ℝ) : D · C ≥ 0 for each C ∈ \overline{NE}_{1}(X)\}.

These cones are governed by the following general results:

Theorem 2.2. Let X be a proper variety and D a Cartier divisor on X.

(1) (Nakai criterion) D is ample if and only if D^{dim(Z)} ·Z > 0 for each closed subvariety Z ⊂ X.

(2) (Kleiman criterion) Assume X is projective. Then D is ample if and only if D ∈ NM_{∘}^{1}(X), i.e., D · C > 0 for each nonzero class C in the closure of the cone of curves on X.

It is not difficult to verify that an ample divisor necessarily satisfies these conditions; we leave this as an exercise. We refer the reader to [KM98, §1.5] for proofs that these conditions are sufficient in general and to [Har77, §V.1] for Nakai’s criterion in the special case of smooth projective surfaces.

Theorem 2.3. Let X be a smooth projective variety. A divisor D ∈ N^{1}(X, ℤ) lies in the pseudoeffective cone \overline{NE}^{1}(X) if and only if, for each ample H and rational ϵ > 0, some multiple of D + ϵH is effective. It lies in the interior

NE_{∘}^{1}(X) ⊂ \overline{NE}^{1}(X)

if and only if there exists an N ≫ 0 so that

ND = A + E

where A is ample and E is effective. Such divisors are said to be big.

Proof. First, any divisor of the form H + E is in the interior of the pseudoeffective cone. If B is an arbitrary divisor then nH +B is very ample for some n > 0, and n(H + E) + B is effective.

Conversely, let D lie in the interior of \overline{NE}^{1}(X). Consider

D − NM_{∘}^{1}(X) ⊂ N^{1}(X, ℝ),

i.e., the cone of anti-ample divisor classes translated so that the vertex is at D. Note that the ample cone of X is open, so we can pick a

B ∈ N^{1}(X, ℚ) ∩ (D − NM_{∘}^{1}(X))

and m > 0 so that E := mB is an effective divisor. Express

B = D − tA

for A ample and t ∈ ℚ_{>0}. Thus

D = \frac{1}{m}E + tA

and clearing denominators gives the desired result.

If D is not in \overline{NE}^{1}(X) then, for each H ample, there exists an ϵ > 0 so that D + ϵH is not effective. Conversely, if D ∈ \overline{NE}^{1}(X) then D + ϵH ∈ NE_{∘}^{1}(X) and we can write

N(D + ϵH) = A + E

where N ≫ 0, A is ample, and E is effective. □

Corollary 2.4. Let X be a smooth projective surface. A nef divisor D is big if and only if D^{2} > 0. Indeed, any divisor D (not necessarily nef) such that D^{2} > 0 and D · H > 0 for some ample divisor H is big.

In fact, the analogous statement is true in all dimensions [KM98, 2.61].

Proof. If D is big then we can express D = A + E, where A is an ample ℚ-divisor and E is an effective ℚ-divisor. We expand

D^{2} = D · (A + E) ≥ D · A = A · A + A · E > 0.

Conversely, if D^{2} > 0 then the Riemann-Roch formula implies either

h^{0}(O_{X}(mD)) ≥ \frac{D^{2}}{2} m^{2}, m ≫ 0,

or

h^{2}(O_{X}(mD)) ≥ \frac{D^{2}}{2} m^{2}, m ≫ 0.

The latter possibility would imply K_{X} − mD is effective for m ≫ 0, which is incompatible with D being nef. (Actually, we only need that D · H > 0 for some ample divisor H.) Given A very ample, a straightforward dimension count shows that h^{0}(O_{X}(mD − A)) remains positive for m ≫ 0, i.e., that D can be expressed as a sum of an ample and an effective divisor. □

Exercises.

Exercise 2.1.1. Let X denote the blow-up of ℙ^{2} at a point. Give examples of big divisors D on X with D^{2} < 0.

Exercise 2.1.2. Let X be a smooth projective variety. Verify that the conditions of the Nakai and Kleiman criteria are necessary for a divisor to be ample. When X is a surface, deduce the sufficiency of the Kleiman criterion from the Nakai criterion.

Exercise 2.1.3. The volume of a Cartier divisor D on an n-dimensional projective variety X is defined [Laz04, 2.2.31] as

vol(D) = lim sup_{m→∞} h^{0}(X, O_{X}(mD))/(m^{n}/n!).

When X is a smooth surface, show that D is big if and only if vol(D) > 0.

2.2. Examples of effective cones of surfaces. For surfaces that are obtained by blowing up the plane β : X → ℙ^{2}, we write L for the pullback of the line class on ℙ^{2} and E_{1}, E_{2}, … for the exceptional curves.

(1) Let f_{1} and f_{2} denote the rulings of X = ℙ^{1} × ℙ^{1}, so that Pic(X) = ℤf_{1} + ℤf_{2}. Then we have

\overline{NE}_{1}(X) = \{a_{1}f_{1} + a_{2}f_{2} : a_{1}, a_{2} ≥ 0\}

and

NE_{1}^{∘}(X) = {a_{1}f_{1} + a_{2}f_{2} : a_{1}, a_{2} > 0}.

(2) If X is ℙ^{2} blown up at one point then

\overline{NE}_{1}(X) = \{aE + b(L − E) : a, b ≥ 0\}.

(3) If X is ℙ^{2} blown up at two points then

\overline{NE}_{1}(X) = \{a_{1}E_{1} + a_{2}E_{2} + b(L − E_{1} − E_{2}) : a_{1}, a_{2}, b ≥ 0\}.

(4) If X is ℙ^{2} blown up at three non-collinear points then

\overline{NE}_{1}(X) = ⟨L − E_{1} − E_{2}, L − E_{1} − E_{3}, L − E_{2} − E_{3}, E_{1}, E_{2}, E_{3}⟩ ,

i.e., the cone generated by the designated divisors.

(5) If X is a cubic surface with lines ℓ_{1}, …, ℓ_{27} then

\overline{NE}_{1}(X) = ⟨ℓ_{1}, …, ℓ_{27}⟩ .

We describe a technique for verifying these claims, using the crucial fact that curves and divisors coincide on surfaces, i.e., \overline{NE}_{1}(X) = \overline{NE}^{1}(X). To decide whether a collection of irreducible curves Γ = \{C_{1}, …, C_{N}\} generates \overline{NE}_{1}(X), it suffices to

• Compute a set of generators Ξ for the dual cone ⟨Γ⟩^{∗}; there are computer programs like PORTA [CL97] and Polymake [GJ00] which can extract Ξ from Γ.

• Check that each A_{i} ∈ Ξ can be written A_{i} = ∑ m_{ij}C_{j}, m_{ij} ≥ 0. Here is why this works: If D is effective then we can write

D = M + F, F = ∑_{j} n_{j}C_{j}, n_{j} ≥ 0,

where M is effective with no support at C_{1}, …, C_{N}. (Here F is the portion of the fixed part of D supported in Γ.) In particular, M ·C_{j} ≥ 0 for each j, i.e., M ∈ ⟨Γ⟩^{∗}. But then M = ∑_{i} a_{i}A_{i} with a_{i} ≥ 0. Thus we have

M = ∑_{ij} a_{i}m_{ij}C_{j}

and D is an effective sum of the C_{j}.

Example 2.5. For X = Bl_{p_{1},p_{2}}ℙ^{2} take Γ = {E_{1}, E_{2}, L − E_{1} − E_{2}}, which generates a simplicial cone. The dual generators are

Ξ = {L − E_{1}, L − E_{2}, L}

and we can write

\begin{matrix}L − E_{1} = (L − E_{1} − E_{2}) + E_{2}, L − E_{2} = (L − E_{1} − E_{2}) + E_{1}, \\ L = (L − E_{1} − E_{2}) + E_{1} + E_{2}. \end{matrix}

It follows that Γ generates \overline{NE}_{1}(X).

Exercises.

Exercise 2.2.1. Verify each of our claims about the generators of the effective cone.

2.3. Extremal rays. We’ll need the following general notion from convex geometry:

Definition 2.6. Given a closed cone C ⊂ ℝ^{n}, an element R ∈ C generates an extremal ray if for each representation

R = D_{1} + D_{2}, D_{1}, D_{2} ∈ C

we have D_{1}, D_{2} ∈ ℝ_{≥0}R.

We will conflate the element R and the ray ℝ_{≥0}R. For a polyhedral cone, i.e., one generated by a finite number of elements

C = ⟨C_{1}, …, C_{N}⟩ = ℝ_{≥0}C_{1} + ⋯ + ℝ_{≥0}C_{N},

the extremal rays correspond to the irredundant generators. On the other hand, for the cone over the unit circle

{(x, y, z) : x^{2} + y^{2} − z^{2} ≤ 0} ⊂ ℝ^{3}

each point of the circle yields an extremal ray.

Our main examples of extremal rays are (−1)-curves:

Proposition 2.7. Let X be a smooth projective surface and E a (−1)-curve. Then E is extremal in \overline{NE}_{1}(X).

If β : X → Y is the blow-down of E then

β_{∗}\overline{NE}_{1}(X) = \overline{NE}_{1}(Y),

hence faces of \overline{NE}_{1}(Y) correspond to faces of \overline{NE}_{1}(X) containing E.

Proof. If we could express E = D_{1} +D_{2} with D_{1}, D_{2} ∈ \overline{NE}_{1}(X) not in ℝ_{≥0}E, then

0 = β_{∗}D_{1} + β_{∗}D_{2}

for nonzero β_{∗}D_{i} ∈ \overline{NE}_{1}(Y). This contradicts the fact that \overline{NE}_{1}(Y) is strongly convex, i.e., that the origin is extremal.

The inclusion

β_{∗}\overline{NE}_{1}(X) ⊂ \overline{NE}_{1}(Y)

is clear because the image of an effective divisor is effective. On the other hand, suppose that D is effective on Y. Since Y is nonsingular, D is a Cartier divisor and β^{∗}D is a well-defined effective Cartier divisor. The projection formula β_{∗}β^{∗}D = D shows that D ∈ β_{∗}\overline{NE}_{1}(X). □

Corollary 2.8. Let X be Del Pezzo and β : X → Y a blow-down morphism. Then Y is Del Pezzo.

Proof. Indeed, we have the discrepancy formula

β^{∗}K_{Y} = K_{X} − E,

where E is the exceptional curve. Since −K_{X} is positive on \overline{NE}_{1}(X) \ \{0\}, −K_{Y} is positive on \overline{NE}_{1}(Y) \ \{0\}. A direct argument that −K_{Y} is ample can be extracted from the proof of the Castelnuovo Contraction Criterion (Theorem 1.12). □

Definition 2.9. Let X be a smooth projective surface. The positive cone C denotes the component of

{D : D^{2} > 0} ⊂ N^{1}(X, ℝ)

containing the hyperplane class. (The Hodge index theorem implies this has two connected components.) Let \overline{C} denote its closure.

We can formulate a more general version of Proposition 2.7, which complements Theorem 2.3 and Corollary 2.4:

Proposition 2.10. [Kol96, II.4] Let X be a smooth projective surface. Then each irreducible curve D with D^{2} ≤ 0 lies in the boundary ∂\overline{NE}_{1}(X). Furthermore,

(2.1)  \overline{NE}_{1}(X) = \overline{C} + ∑_{D} ℝ_{≥0}D

where the sum is taken over irreducible curves D with D^{2} < 0.

Proof. Corollary 2.4 implies \overline{NE}_{1}(X) ⊃ \overline{C}, and \overline{NE}_{1}(X) ⊃ \overline{C} + ∑_{D} ℝ_{≥0}D follows immediately. It remains to establish the reverse inclusion.

First, suppose that D is irreducible with D^{2} ≤ 0. Let H be a very ample divisor of X. Then for each rational ϵ > 0 we claim that D − ϵH fails to be in the effective cone. Indeed, if D − ϵH were effective then it could be expressed as a nonnegative linear combination of irreducible curves, some different from D,

D − ϵH ≡ c_{0}D + ∑_{j} c_{j}D_{j}, c_{0} ∈ [0, 1), c_{j} > 0.

Regrouping terms, for some ϵ′ > 0 we obtain

D − ϵ′H ≡ ∑_{j} c′_{j}D_{j}, c′_{j} > 0.

However, this contradicts the fact that

D · (D − ϵ′H) < 0.

For D irreducible with D^{2} < 0 consider the closed cone

V = ⟨z ∈ \overline{NE}_{1}(X) : z · D ≥ 0⟩ ⊂ \overline{NE}_{1}(X),

which contains all effective divisors without support along D. Thus \overline{NE}_{1}(X) is the smallest convex cone containing V and D. Since D ∉ V, it follows that D is extremal in \overline{NE}_{1}(X).

Conversely, suppose that Z is extremal with Z^{2} < 0. There is necessarily some irreducible curve C such that C · Z < 0. Let Z_{i} denote a sequence of effective ℚ-divisors approaching Z. Since Z_{i} · C < 0 for i ≫ 0, we must have that C^{2} < 0. Moreover C appears in Z_{i} with coefficient c_{i} and c = lim c_{i} > 0. Thus Z − cC is pseudoeffective and C ∈ ℝ_{≥0}Z by extremality. □

For special classes of surfaces, the negative extremal rays are necessarily (−1)- curves or (−2)-curves, i.e., smooth rational curves with self-intersection −2:

Corollary 2.11. [Kol96, II.4.14] Suppose X is a smooth projective surface with −K_{X} nef. Then the sum in expression (2.1) can be taken over D with D^{2} = −1 or −2 and D ≃ ℙ^{1}.

Proof. Since K_{X} · D ≤ 0 and D^{2} < 0 then the adjunction formula

2g(D) − 2 = D^{2} + K_{X} · D

allows only the possibilities listed. □

Exercises.

Exercise 2.3.1. Classify extremal rays and describe decomposition (2.1) for: • ℙ^{2} blown up at two points or three non-collinear points;

• the Hirzebruch surfaces 𝔽_{d}.

2.4. Structural results on the cone of curves I. The closed cone of effective curves has a very nice structure in the region where the canonical class fails to be nef. There are two different approaches to these structural results. The first emphasizes vanishing theorems (for higher cohomology) on line bundles and the resulting implications for linear series, e.g., basepoint-freeness. You can find details of this approach in references such as [Rei97, D] and [CKM88]. One significant disadvantage is that the reliance on Kodaira-type vanishing makes generalization to positive characteristic problematic. The second approach emphasizes the geometric properties of the curves themselves, especially the bend-and-break technique of Mori. This approach is taken in Mori’s original papers, as well as in [Kol96, II.4,III.1].

Since both approaches are important for applications, we will sketch the key elements of each, referring to the literature for complete arguments.

Theorem 2.12 (Cone Theorem). [Rei97, D.3.2] [KM98, Thm. 3.7] [CKM88, 4.7] Let X be a smooth projective surface with canonical class K_{X}. There exists a countable collection of R_{i} ∈ \overline{NE}_{1}(X) ∩ N_{1}(X, ℤ) with K_{X} · R_{i} < 0 such that

\overline{NE}_{1}(X) = \overline{NE}_{1}(X)_{K_{X}≥0} + ∑_{i} ℝ_{≥0}R_{i},

where the first term is the intersection of \overline{NE}_{1}(X) with the halfplane \{v ∈ N_{1}(X, ℝ) : v · K_{X} ≥ 0}. Given any ample divisor H and ϵ > 0, there exists a finite number of R_{i} satisfying (K_{X} + ϵH) · R_{i} ≤ 0.

Corollary 2.13. Let X be a Del Pezzo surface. Then \overline{NE}_{1}(X) is a finite rational polyhedral cone.

What’s even more remarkable is that the extremal rays can be interpreted geometrically. The following theorem should be understood as a far-reaching extension of the Castelnuovo contraction criterion (Theorem 1.12):

Theorem 2.14 (Contraction Theorem). [Rei97, D.4] [KM98, Theorem 3.7] Let X be a smooth projective surface and R a generator of an extremal ray with K_{X} · R < 0. There exists a morphism ϕ : X → Y to a smooth projective variety, with the following properties:

(1) ϕ_{∗}R = 0 and ϕ contracts those curves with classes in the ray ℝ_{≥0}R;

(2) ϕ has relative Picard rank one and Pic(Y) can be identified with R^{⊥} ⊂ Pic(X).

Proof. The proofs of Theorems 2.12 and 2.14 are intertwined. We can only offer a sketch of the arguments required. Some of these work only in characteristic zero, but we will make clear which ones.

Suppose we want to analyze the part of the effective cone along which K_{X} is negative. Fix an ample divisor H, which is necessarily positive along \overline{NE}_{1}(X). Which divisors τK_{X} + H, τ ∈ [0, 1], are nef? Consider the nef threshold

t = sup{τ ∈ ℝ : τK_{X} + H nef },

i.e., (tK_{X}+H)^{⊥} is a supporting hyperplane of \overline{NE}_{1}(X) provided tK_{X}+H is nonzero. If we choose H suitably general, we can assume this hyperplane meets \overline{NE}_{1}(X) in an extremal ray. (Of course, for special H it might cut out a higher-dimensional face.)

Figure 1. Finding a supporting hyperplane of the cone of curves (drawn in the projectivization of N_{1}(X, ℝ))

The first step is the rationality of the nef threshold. The most straightforward proof [Rei97, D.3.1] uses the Riemann-Roch formula and Kodaira vanishing, and thus is valid only in characteristic zero:

Lemma 2.15 (Rationality). The nef threshold is rational.

Thus the K_{X}-negative extremal rays of the cone of effective curves are determined by linear inequalities with rational coefficients. These rays here can be chosen to be integral.

The second step is to show that the (ℚ-)divisor D := tK_{X} + H is semiample. Then the resulting morphism ϕ : X → Y will contract precisely the extremal rays in the face supported by the hyperplane (tK_{X} + H)^{⊥}, which gives the contraction theorem.

Lemma 2.16 (Basepoint-freeness). Let D be a nef ℚ-divisor such that D = tK_{X} + H for H ample and t > 0. Then D is semiample.

Proof. Since D is nef we have D^{2} ≥ 0. If D^{2} > 0 then the Nakai criterion (Theorem 2.2) implies D is ample unless there exists an irreducible curve E with D · E = 0. The Hodge index theorem implies E^{2} < 0. Since K_{X} · E < 0, Proposition 1.6 implies that E is a (−1)-curve. The desired contraction exists by the Castelnuovo Criterion (Theorem 1.12).

Now suppose D^{2} = 0. If D is numerically equivalent to zero then −K_{X} is ample. In this situation, D is semiample if and only if it is torsion, which is a consequence of the following lemma.

Lemma 2.17. Suppose that X is a smooth projective surface and −K_{X} is nef and big, e.g., X is a Del Pezzo surface. Then

• H^{2}(X, O_{X}) = 0 and Pic(X) is smooth;

• H^{1}(X, O_{X}) = 0 and the identity component of Pic(X) is trivial.

Proof. Since some positive multiple of −K_{X} is effective, no positive multiple of K_{X} is effective. Thus

h^{2}(X, O_{X}) = h^{0}(X, O_{X}(K_{X})) = 0

and Pic(X) is smooth. The identity component has dimension q = h^{1}(X, O_{X}).

Let b_{i}(X) denote the ith Betti number of X; in positive characteristic, we define these using étale cohomology [Mil80]. Recall the formulas [Mil80, III.4.18,V.3.12] b_{1}(X) = 2q and

c_{2}(X) = χ(X) = b_{0}(X) − b_{1}(X) + b_{2}(X) − b_{3}(X) + b_{4}(X) = 2 − 4q + b_{2}(X).

Noether’s formula

12χ(O_{X}) = c_{1}(X)^{2} + c_{2}(X)

and the fact that c_{1}(X)^{2} = K_{X}^{2} > 0 imply

12(1 − q) > 2 − 4q + b_{2}(X).

Consequently

10 > 8q + b_{2}(X)

and thus q = 0 or q = b_{2}(X) = 1. To exclude the last case, observe that if the identity component of the Picard group is positive dimensional then so is the Albanese variety. (Indeed, these abelian varieties are dual to each other.) Furthermore, the Albanese map X → Alb(X) [Lan59, II.3] is a dominating morphism to an elliptic curve. The classes of a fiber and the pullback of an ample divisor from the Albanese are necessarily independent; thus b_{2}(X) ≥ 2. □

We return to the proof of Lemma 2.16. If D is not numerically trivial then K_{X} · D < 0 and Riemann-Roch imply that h^{0}(X, O_{X}(mD)) grows at least linearly in m. And since Corollary 2.4 ensures that D is not big, h^{0}(O_{X}(mD)) cannot be a quadratic function of m. Decompose D into a moving and a fixed part

D = M + F, M^{2} ≥ 0, M · F ≥ 0.

Note that D · F = M · F + F^{2} ≥ 0 (since D is nef), M^{2} = 0 (as M is not big), and F^{2} ≤ 0 (since F is not big). On the other hand,

0 = D^{2} = 2M · F + F^{2} ≥ M · F

so M · F = 0 and F^{2} = 0 as well. The Hodge index theorem implies that M and F are proportional in the Néron-Severi group, provided they are numerically nontrivial. In particular, if F ≠ 0 then K_{X} · F < 0 and h^{0}(F, O_{X}(mF)) grows linearly in m, contradicting the fact that F is fixed. Thus D = M is moving with perhaps isolated basepoints, the number of which is bounded by M^{2} = 0. We conclude that D is basepoint-free. □

This completes the proof of Theorem 2.14.

Remark 2.18. This argument yields another result we shall use later: Let X be a smooth projective surface with −K_{X} nef and big. Assume that D is a nef line bundle on X with D^{2} = 0. Then D is semiample.

The third step is to bound the denominator of the nef threshold (cf. [Rei97, D.3.1] and [CKM88, 12.12]):

Lemma 2.19 (Bounding denominators). Assume the nef threshold is rational. Then its denominator is ≤ 3.

Proof. Again, the argument is a case-by-case analysis of D = tK_{X} + H. If D^{2} > 0 then D is orthogonal to a (−1)-curve and t ∈ ℤ. If D is numerically trivial then −K_{X} is ample and Lemma 2.17 implies χ(O_{X}) = 1. Express −K_{X} = rL, where L is a primitive ample divisor and r ∈ ℕ. It suffices to show that r ≤ 3. Noether’s formula and the argument for Lemma 2.17 give

12 = r^{2}L^{2} + c_{2}(X) = r^{2}L^{2} + 2 + b_{2}(X)

so the only possibilities are r = 1, 2, 3.

It remains to consider the situation where D^{2} = 0 but D ≢ 0. As we’ve seen, D · K_{X} < 0 in this case. Furthermore, t′K_{X} + H is never effective for t′ > t. Here we claim 2t ∈ ℕ. Otherwise, there exist m, n ∈ ℕ with m ≫ 0 such that

mt = n + α, 1/2 < α < 1.

Thus nK_{X} + mH is ample, Γ(O_{X}(−nK_{X} − mH)) = 0, and

H^{2}(O_{X}((n + 1)K_{X} + mH)) = 0

by Serre duality. We deduce

h^{0}(O_{X}((n + 1)K_{X} + mH)) = χ(O_{X}) + \frac{1}{2}((n + 1)K_{X} + mH) · (nK_{X} + mH)

= χ(O_{X}) + \frac{1}{2}(−α(1 − α)K_{X}^{2} + m(1 − 2α)D · K_{X} + m^{2}D^{2}) = constant + m · positive number

which is positive for m sufficiently large. Thus mH + (n + 1)K_{X} is effective, a contradiction. □

To complete the proof of Theorem 2.12, we show that the K_{X}-negative extremal rays have no accumulation points and for any ample H there are finitely many such rays in the region (K_{X} + ϵH) ≤ 0. Let H_{1}, …, H_{d} denote ample divisors forming a basis for the Néron-Severi group such that

H = a_{1}H_{1} + ⋯ + a_{d}H_{d}, a_{1}, …, a_{d} ∈ ℚ_{>0}.

Let t_{j} denote the nef threshold of H_{j}. Consider the local coordinate functions

b_{j}(γ) = \frac{H_{j} · γ}{−K_{X} · γ}

on the open subset of ℙ(N_{1}(X, ℝ)) where K_{X} ≠ 0. For K_{X}-negative extremal rays, b_{j} ≥ t_{j}; Lemmas 2.15 and 2.19 imply these are rational numbers with denominators dividing six. It follows that these rays have no accumulation points. The extremal rays with (NK_{X} + H) · R_{i} ≤ 0 for some N ∈ ℕ have coordinates satisfying

a_{1}b_{1} + ⋯ + a_{d}b_{d} ≤ N.

Since the a_{j} and b_{j} are positive rational numbers with bounded denominators, there are at most finitely many possibilities. □

The structure of the contraction morphism ϕ : X → Y depends on the intersection properties of irreducible curves E generating our extremal ray: Case E^{2} < 0: Proposition 1.6 ensures E is a (−1)-curve and ϕ is the blow-down of E. Case E^{2} = 0: By adjunction, E ≃ ℙ^{1} and ϕ : X → Y fibers X over a curve with generic fiber ℙ^{1}. The extremality implies all fibers are irreducible and reduced, thus every fiber of ϕ is a projective line and we have a minimal ruled surface. Case E^{2} > 0: Corollary 2.4 implies E is big. Since f contracts E and all its deformations, ϕ is constant. Thus Pic(X) = ℤ and X is Del Pezzo.

Since the first case cannot occur when X is minimal, we obtain:

Corollary 2.20. Let X be a minimal smooth projective surface. Then one of the following conditions holds:

• K_{X} is nef;

• X is a ℙ^{1}-bundle over a curve B;

• X is Del Pezzo with Pic(X) = ℤ.

In the first instance, X is the unique minimal smooth projective surface in its birational equivalence class.

Proof. We only have to establish uniqueness: Let X′ be another minimal smooth projective surface birational to X. Choose a factorization

Y

↙^{ϕ′}  ↘^{ϕ}
X′  X

where Y is smooth projective and the morphisms are birational. Indeed, ϕ and ϕ′ are sequences of contractions of (−1)-curves (see Theorem 1.13). Express K_{Y} = ϕ^{∗}K_{X} + F where F is an effective divisor with support equal to the exceptional locus of ϕ. If E ⊂ Y is a (−1)-curve contracted by ϕ′ then

−1 = K_{Y} · E ≥ F · E

as K_{X} is nef. Thus E is contained in the support of F and is contracted by ϕ′. Since each ϕ′-exceptional divisor is ϕ-exceptional, we have a factorization

ϕ : Y → X′ → X.

Since X′ is minimal, it must equal X. □

Remark 2.21 (Relative version). Given a morphism f : X → B to a variety, we can also consider the relative cone of effective curves

\overline{NE}_{1}(f : X → B) = \{D ∈ \overline{NE}_{1}(X) : f_{∗}D = 0\}.

When B is smooth, this is the intersection of \overline{NE}_{1}(X) with the orthogonal complement to f^{∗}Pic(B). The Cone Theorem 2.12 describes its structure in the region where the canonical divisor K_{X} is negative. There is a relative version of the Contraction Theorem giving contractions over B

\begin{aligned}X & & ⟶^{ϕ} & & Y \\ & ↘ & & ↙ \\ & & B \end{aligned} .

The classification of ruled surfaces (Theorem 1.16) is a prime example.

Exercises.

Exercise 2.4.1. Let X be a smooth projective surface with K_{X} nef. Show that X is not rational.

Exercise 2.4.2. Let p_{1}, …, p_{8} ∈ ℙ^{2} be general points. Let p_{0} be the last basepoint of the pencil of cubic curves containing these points. Show that X = Bl_{p_{0},p_{1},...,p_{8}}ℙ^{2} has an infinite number of (−1)-curves.

Hints: Let E_{0}, …, E_{8} denote the exceptional divisors. Consider the elliptic fibration

η : X → ℙ^{1}

induced by the linear series |f| with f = −K_{X} = 3L − E_{0} − ⋯ − E_{8}. Verify that sections of η are all (−1)-curves. Designate E_{0} as the zero section of η and let σ_{i} : ℙ^{1} → X, i = 1, …, 8 denote the sections associated with E_{1}, …, E_{8}. Given sections σ, σ′ : ℙ^{1} → X we have

\begin{aligned}[(σ + σ′)(ℙ^{1})] & = & [σ(ℙ^{1})] + [σ′(ℙ^{1})] − [E_{0}] + w(σ, σ′)[f] \\ & & w(σ, σ′) = −[σ(ℙ^{1}) − E_{0}] · [σ′(ℙ^{1}) − E_{0}]. \end{aligned}

Use this to analyze m_{1}σ_{1} + ⋯ + m_{8}σ_{8}.

2.5. Structural results on the cone of curves II. Our discussion of the Cone and Contraction Theorems is missing one crucial element: We have not shown that the extremal rays are generated by classes of rational curves on X. Another issue is that we cited arguments for the rationality of extremal rays relying on vanishing theorems; these do not readily extend to positive characteristic. These gaps can be filled using Mori’s ‘bend-and-break’ technique:

Theorem 2.22 (Bend-and-break). [Kol96, II.5.14] Let X be a smooth projective variety, C a smooth projective curve, and f : C → X a morphism. Let M be a nef ℝ-divisor. Assume that −K_{X} · C > 0. Then for each x ∈ f(C) there is a rational curve x ∈ D_{x} ⊂ X such that

M · D_{x} ≤ 2 dim(X)\frac{M · C}{−K_{X} · C} , −K_{X} · D_{x} ≤ dim(X) + 1.

This is a deep result that we will not prove here. The main idea is to use the fact that the anticanonical class is negative to show that f admits deformations f_{t} : C → X whose images still contain x. This strategy works beautifully provided C has genus zero, but in higher genus it is necessary to reduce modulo p and precompose f with the Frobenius map. Then we consider limits of f_{t}(C) ⊂ X as t → 0; these necessarily contain rational curves x ∈ D′ ⊂ X. We can iterate this strategy until we obtain a rational curve D_{x} ∋ x with fairly small anticanonical degree, i.e., −K_{X} · D_{x} ≤ dim(X) + 1.

We still have not mentioned the rôle of the divisor M. This is crucial in applications to the cone of curves:

Theorem 2.23 (Cone Theorem bis). [Kol96, III.1.2] Let X be a smooth projective surface with canonical class K_{X}. There exists a countable collection of rational curves L_{i} ⊂ X with 0 < −K_{X} · L_{i} ≤ 3 such that

(2.2)  \overline{NE}_{1}(X) = \overline{NE}_{1}(X)_{K_{X}≥0} + ∑_{i} ℝ_{≥0}[L_{i}].

Given any ample divisor H and ϵ > 0, there exists a finite number of L_{i} satisfying (K_{X} + ϵH) · L_{i} ≤ 0.

Proof. We offer a sketch proof following [Kol96]: Let M be an ℝ-divisor corresponding to a supporting hyperplane of a K_{X}-negative extremal ray R ∈ \overline{NE}_{1}(X). Thus M · R = 0 and M · γ > 0 for γ ∈ \overline{NE}_{1}(X) with γ ∉ ℝ_{≥0}R; in particular, M is a nef ℝ-divisor. We show that M is a supporting hyperplane of the closure W of the cone associated to the right-hand side of (2.2). (We refer the reader to [Kol96, III.1] for the argument that the right-hand side of (2.2) defines a closed cone.)

Assume this is not the case. Rescaling M if necessary, we may assume that M ·D ≥ 1. for each irreducible curve D ⊂ X with [D] ∈ W. Consider the functional

\begin{matrix}ϕ : N_{1}(X, ℝ)_{K_{X}<0} & → & ℝ \\ γ & ↦ & −M · γ/K_{X} · γ \end{matrix}

which is nonnegative on \overline{NE}_{1}(X)_{K_{X}<0} and positive away from ℝ_{≥0}R. Choose a sequence of effective curves with real coefficients approaching R

C_{i} = ∑_{j} a_{ij}C_{ij}, a_{ij} > 0, lim_{i→∞} C_{i} = R.

For each i, there exists an index j such that K_{X} · C_{ij} < 0 and ϕ(C_{ij}) ≤ ϕ(C_{i}). And we have lim_{i→∞} ϕ(C_{i}) = ϕ(R) = 0.

On the other hand, bend-and-break yields rational curves L_{i} such that −K_{X} · L_{i} ≤ 3 and

M · L_{i} ≤ 4ϕ(C_{ij}) < 4ϕ(C_{i}).

The left-hand side is bounded from below by 1 while the right-hand side approaches zero, so we obtain a contradiction. □

2.6. Classification of surfaces.

Theorem 2.24. If X is a Del Pezzo surface with Pic(X) = ℤ then X ≃ ℙ^{2}.

Proof. Our argument follows [Kol96, III.3.7]. We first offer a short proof in characteristic zero: Let L be a line bundle generating Pic(X) with L · K_{X} < 0. Lemma 2.17 ensures that H^{1}(O_{X}) = H^{2}(O_{X}) = 0, so by Hodge theory we have b_{1}(X) = 0 and b_{2}(X) = 1. Poincaré duality then implies L · L = 1. Noether’s formula

12χ(O_{X}) = c_{1}^{2}(X) + c_{2}(X)

implies c_{1}^{2}(X) = K_{X}^{2} = 9. We conclude that K_{X} = −3L and χ(L) = 3. Since h^{2}(X, L) = h^{0}(X, K_{X} − L) = 0, we have h^{0}(X, L) ≥ 3. Moreover, the members of the corresponding linear series are integral curves of genus zero, i.e., ℙ^{1}’s. A straightforward inductive argument shows that L is basepoint-free and thus induces a degree-one morphism X → ℙ^{2}, i.e., X ≃ ℙ^{2}.

We only used characteristic zero to show that K_{X} = −3L. Suppose then that K_{X} = −rL for some r ∈ ℕ, where L is a generator of Pic(X). We have already seen in the proof of Lemma 2.19 that r = 1, 2, 3. If r = 2 then

2g(L) − 2 = L^{2} + K_{X}L = −L^{2}

so L^{2} = 2 and g(L) = 0; Riemann-Roch then gives χ(X, L) = 4. Arguing as above, L is basepoint-free and defines a morphism ϕ : X → ℙ^{n} for n ≥ 3. The image is a quadric surface or a plane, and the latter possibility would contradict nondegeneracy. However, a quadric surface cannot be a rank-one Del Pezzo surface.

Finally, suppose that r = 1. The Cone Theorem (Theorem 2.23) implies the existence of a rational curve f : ℙ^{1} → X with deg f^{∗}(−K_{X}) = deg f^{∗}L ≤ 3. We have f_{∗}[ℙ^{1}] = mL for some m ∈ ℕ with mL^{2} ≤ 3, and consequently K_{X}^{2} ≤ 3. Since every curve in X has positive self-intersection, we can deduce a contradiction from the following fact:

Lemma 2.25. Let Y be a Del Pezzo surface with K_{Y}^{2} ≤ 4. Then Y contains a (−1)-curve.

Such curves C are called lines because −K_{Y} · C = 1.

There are two general approaches to this. The most direct (see [Kol96, III.3.6] or Exercise 2.6.2) is to express Y as a hypersurface in a suitable weighted projective space, i.e., as a cubic surface in ℙ^{3} (when K_{Y}^{2} = 3), a quartic surface in ℙ(1, 1, 1, 2) (when K_{Y}^{2} = 2), or as a sextic surface in ℙ(1, 1, 2, 3) (when K_{Y}^{2} = 1). Proving this entails a fair amount of ad hoc analysis of linear series. Another approach (cf. [Isk79]) involves showing that Y lifts to characteristic zero and using the classification tools available there. □

Theorem 2.26 (Castelnuovo’s Criterion). [Bea96, V.6] [Kol96, III.2.4] Let X be a smooth projective minimal surface. Then X is rational if and only if

q(X) = h^{1}(O_{X}) = 0, P_{2}(X) := h^{0}(X, O_{X}(2K_{X})) = 0.

Proof. The necessity of the numerical conditions is clear, as P_{2}(X) and q(X) are birational invariants of smooth projective varieties. For sufficiency, we may assume that X is minimal and falls into one of the three categories of Proposition 2.20. The third case (where X is Del Pezzo with Pic(X) = ℤ) is rational by Theorem 2.24. In the second case (where X is ruled over a curve B), the assumption q(X) = 0 implies that B has genus zero. Corollary 1.22 yields that X is rational. Finally, suppose that K_{X} is nef, so in particular K_{X}^{2} ≥ 0. We know that K_{X} is not effective; if Γ(X, O_{X}(K_{X})) ≠ 0 then Γ(X, O_{X}(2K_{X})) ≠ 0. Thus

χ(O_{X}) = h^{0}(O_{X}) − h^{1}(O_{X}) + h^{2}(O_{X}) = 1

and

χ(O_{X}(−K_{X})) = K_{X}^{2} + 1 ≥ 1.

Since h^{2}(O_{X}(−K_{X})) = h^{0}(O_{X}(2K_{X})) = 0 we conclude h^{0}(O_{X}(−K_{X})) > 0, i.e., −K_{X} is effective. As K_{X} is nef, the only possibility is K_{X} trivial, a contradiction.

□

Corollary 2.27. Del Pezzo surfaces are rational.

Proof. Let X be a Del Pezzo surface. Since −K_{X} is ample we have that P_{2}(X) = 0. Lemma 2.17 gives h^{1}(O_{X}) = 0. □

Corollary 2.28. Each Del Pezzo surface X is isomorphic to one of the following:

• ℙ^{1} × ℙ^{1};

• a blow-up of ℙ^{2} at eight or fewer points.

Proof. By Corollary 2.8, we just need to show that minimal Del Pezzo surfaces X are either ℙ^{2} or ℙ^{1} × ℙ^{1}. Our previous analysis implies X is ℙ^{2} or a Hirzebruch surface 𝔽_{d}. But then X contains a rational curve of self-intersection −d, so d = 0, 1 by Proposition 1.6. □

Remark 2.29. The classification of complex surfaces goes back to the work of Castelnuovo and Enriques in the late 19th and early 20th centuries. The extension to positive characteristic is largely due to Zariski, who first proved the Castelnuovo rationality criterion in this context [Zar58a, Zar58b].

Exercises.

Exercise 2.6.1. Suppose that X is a surface such that K_{X} is not nef and Pic(X) has rank at least three. Then X contains a (−1)-curve.

Exercise 2.6.2. Let Y be a Del Pezzo surface with K_{Y}^{2} = 3 (resp. K_{Y}^{2} = 4). Show that −K_{Y} is very ample and the image under |−K_{Y} | is a cubic surface (resp. complete intersection of two quadric hypersurfaces.) Conclude that Y contains a line (cf. Corollary 1.9).

3. Classifying surfaces over non-closed fields

Let k be a perfect field with algebraic closure k̄ and Galois group G = Gal(k̄/k). Let X be a smooth projective surface over k so that

X̄ = Xk̄ = X ×_{Spec(k)} Spec(k̄)

is connected. We use Pic(X) to denote line bundles on X defined over k.

3.1. Minimal surfaces.

Definition 3.1. A smooth projective surface X over k is minimal if any birational morphism over k to a smooth surface

ϕ : X → Y

is an isomorphism.

Theorem 3.2. X is minimal if and only if X̄ admits no Galois-invariant collection of pairwise disjoint (−1)-curves.

Proof. Suppose X is not minimal and admits a birational morphism ϕ : X → Y. By Theorem 1.13, X̄ admits a (−1)-curve E contracted by ϕ. Since ϕ is birational there are only a finite number of such curves, so let E_{1}, …, E_{r} denote the curves in the Galois orbit of E. As we saw in the proof of Theorem 1.13, the intersection form on ℤE_{1} + ⋯ + ℤE_{r} is negative definite, thus the matrix

( \begin{matrix}E_{i}^{2} & E_{i}E_{j} \\ E_{i}E_{j} & E_{j}^{2} \end{matrix} ) , i ≠ j,

has positive determinant. It follows that E_{i} · E_{j} < 1, which gives the disjointness.

Conversely, let E_{1}, …, E_{r} denote a Galois-invariant collection of pairwise disjoint (−1)-curves. Let H be an ample divisor on X. Since H · E_{i} = H · E_{j} for each i, j, the divisor

H′ = H + ∑_{j=1}^{r}(H · E_{i})E_{i}

is also Galois-invariant. We just take Y = Proj(⨁_{n≥0} Γ(X, nH′)), as in the proof of Theorem 1.12. □

Remark 3.3 (Galois-invariant classes versus divisors defined over k). Not every element L ∈ Pic(X̄)^{G} comes from a line bundle defined over k. Applying the Hochschild-Serre spectral sequence [Mil80, III.2.20], we find

H^{1}(X, O_{X}^{∗}) = ker (H_{G}^{0}H^{1}(X̄, O∗ ; X̄) ⟶^{d_{2}^{01}} H_{G}^{2}H^{0}(X̄, O∗ ; X̄))

which yields

Pic(X) = ker (Pic(X̄)^{G} ⟶^{d_{2}^{01}} Br(k)) .

Since Br(k) is torsion, some power NL with N > 0 is defined.

On the other hand, when X(k) ≠ ∅ the homomorphism d_{2}^{01} is trivial. Indeed, the spectral sequence shows that the image of d_{2}^{01} lies in the kernel of the homomorphism

s^{∗} : Br(k) → Br(X) = H^{2}(X, O_{X}^{∗})

induced by the structure map s : X → Spec(k). Each rational point x : Spec(k) → X induces x^{∗} : Br(X) → Br(k), a left-inverse of s^{∗}. Thus s^{∗} is injective and d_{2}^{01} is trivial. (See [CTS87] for a comprehensive discussion.)

Example 3.4. Suppose we have a cubic surface X with six conjugate disjoint lines E_{1}, …, E_{6}. Does it follow that X is the blow-up of ℙ^{2} at six conjugate points?

The divisor class −K_{X} + E_{1} + ⋯ + E_{6} = 3L is definitely defined over k. The corresponding linear series gives a morphism

X → Y ⊂ ℙ^{9}

blowing down E_{1}, …, E_{6}; here Ȳ ≃ ℙ2 ; k̄ is embedded via the cubic Veronese embedding. This is an example of a Brauer-Severi variety, i.e., a variety Y such that Ȳ ≃ ℙdim(Y) ; k̄. Moreover, the invariant class

L ∈ Pic(X̄)^{G}

comes from Pic(X) if and only if Y ≃ ℙ_{k}^{2}. A diagram-chase shows that d_{2}^{01}(L) ∈ Br(k) vanishes if and only if [Y] ∈ Br(k) is trivial.

Exercises.

Exercise 3.1.1. Let Y be a Brauer-Severi surface. Show there exists a smooth cubic surface X admitting a birational morphism ϕ : X → Y. Hint: A generic vector field on Y vanishes at three Galois-conjugate points. Blow up along two such collections of points.

Exercise 3.1.2 (Degree seven Del Pezzo surfaces). Let X be a surface such that X̄ ≃ Bl_{p_{1},p_{2}}(ℙ^{2}). Show there exists a birational morphism X → ℙ^{2}, obtained by blowing up a pair of Galois-conjugate points.

Exercise 3.1.3 (Some degree eight Del Pezzo surfaces). Let X be a surface such that X̄ ≃ Bl_{p}(ℙ^{2}). Show that X is isomorphic to Bl_{p}(ℙ^{2}) over k.

Exercise 3.1.4 (Degree five Del Pezzo surfaces). [Sko93] [SD72] Let X be a surface such that X̄ ≃ Bl_{p_{1},p_{2},p_{3},p_{4}}(ℙ^{2}), where the points are distinct and no three are collinear.

(1) Show that the four points are projectively equivalent to

[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]

over k̄.

(2) Show that sections of −K_{X} embed X as a quintic surface in ℙ^{5}.

(3) Show that this surface is cut out by five quadrics. Hint: It suffices to verify this on passage to k̄.

(4) Choose generic Q_{0}, Q_{1}, Q_{2} ∈ I_{X}(2). Verify that

Q_{0} ∩ Q_{1} ∩ Q_{2} = X ∪ W,

where W̄ is isomorphic to Bl_{p}ℙ^{2}.

(5) Using Exercise 3.1.3, show that the exceptional divisor E ⊂ W is defined over k and intersects X in one point.

Conclude that X(k) ≠ ∅.

Figure 2. Degenerate fibers of a conic bundle

3.2. Conic bundles. Our treatment owes a great deal to Iskovskikh [Isk79].

Definition 3.5. A conic bundle is a dominant morphism f : X → C from a smooth projective minimal surface X to a smooth curve, so that the generic fiber is a smooth curve of genus zero.

Of course, over an algebraically closed field this is the same as a minimal ruled surface. However, without the Tsen-Lang Theorem we cannot construct a section of f defined over k.

Proposition 1.19 does still apply: It guarantees that each smooth fiber of f is a plane conic and splits over a quadratic extension. It follows that there exists a bisection of f, i.e., an irreducible curve D ⊂ C so that f|_{D} : D → C has degree two. Indeed, intersect the generic fiber (realized as a plane conic) with a line and take the closure in X.

Theorem 3.6. Let f : X → C be a conic bundle. Then any reducible fibers of X̄ → C̄ consist of two (−1)-curves intersecting in one point, conjugate under the Galois action.

Proof. Suppose F is a reducible fiber of X̄; designate the field of definition of f(F) ∈ C by k_{1} ⊃ k. The existence of a reducible fiber guarantees that X̄ is not relatively minimal and F contains a (−1)-curve (Theorem 1.16). Let E_{1}, …, E_{r} be the Galois-orbit under the action of Gal(k̄_{1}/k_{1}); the E_{i} are not pairwise disjoint by minimality (Theorem 3.2).

We claim that the only combinatorial possibility is r = 2, E_{1} · E_{2} = 1, and E_{1}^{2} = E_{2}^{2} = −1. Write T = E_{1} ∪ ⋯ ∪ E_{r} and set n = E_{i} · (∑_{j≠i} E_{j}), i.e., the number of points of intersection of each component with the other components. We can compute the arithmetic genus using

2p_{a}(T) − 2 = −2r + rn.

Since F has arithmetic genus zero p_{a}(T) ≤ 0 and n = 1, i.e., each connected component of T consists of two (−1)-curves meeting at one point. Reordering indices if needed, let E_{1} ∪ E_{2} denote one of these components.

By Proposition 1.18, if E_{1}, …, E_{r+s} are the irreducible components of F then the intersection form on

(⨁_{j=1}^{r+s} ℤE_{j})/ℤF

is negative definite. However, we have (E_{1} + E_{2})^{2} = −1 + 2 − 1 = 0 so necessarily F = E_{1} + E_{2}. This proves the claim and the result. □

We have seen (Proposition 1.19) that the generic fiber of f : X → C admits a natural realization as a smooth plane conic. This is obtained using sections of the dual to the differential one-forms. We can extend this over all of C using the relative dualizing sheaf

ω_{f} = Ω_{X}^{2} ⊗ (f^{∗}Ω_{C}^{1})^{−1}.

We have natural homomorphisms

Ω_{f^{−1}(p)}^{1} → ω_{f^{−1}(p)} = ω_{f}|_{f^{−1}(p)},

where the first arrow is an isomorphism wherever f is smooth.

Corollary 3.7 (Conic bundles really are conic bundles). Let f : X → C be a conic bundle with relative dualizing sheaf ω_{f}. Then we have an embedding over C

X  ↪^{j}  ℙ(f_{∗}ω_{f}^{−1})
↘  ↙
C

realizing each fiber of X as a plane conic.

Proof. We use the classification of fibers in Theorem 3.6. For the smooth fibers, the anticanonical embedding has already been discussed in Proposition 1.19. For the reducible fibers, the anticanonical sheaf is very ample, realizing the fiber as a union of two distinct lines in ℙ^{2}.

Thus for each p ∈ C, ω_{f}^{−1}|_{f^{−1}(p)} is very ample and has no higher cohomology. Cohomology and base change gives that f_{∗}ω_{f}^{−1} is locally free of rank three and has cohomology commuting with base extension. Thus we obtain a closed embedding over C

j : X ↪ ℙ(f_{∗}ω_{f}^{−1})

in a ℙ^{2}-bundle over C. □

Definition 3.8. A rational conic bundle is a conic bundle f : X → C over a curve of genus zero.

3.3. Analysis of Néron-Severi lattices. We analyze the Néron-Severi group of rational conic bundles f : X → ℙ^{1}. Note that K_{X} is defined over the base field.

Theorem 3.6 and Corollary 1.22 imply that X̄ is a blow-up of a Hirzebruch surface at r points in distinct fibers:

\begin{aligned}X̄ & & ⟶ & & 𝔽_{d} \\ & ↘ & & ↙ \\ & & ℙ^{1} \end{aligned} .

The corresponding reducible fibers of X̄ → ℙ^{1} are denoted

E_{1} ∪ E′_{1}, E_{2} ∪ E′_{2}, …, E_{r} ∪ E′_{r},

so that E_{i} + E′_{i} = F for each i.

There are a number of natural lattices to consider. We have the relative Néron-Severi lattice

N^{1}(X̄ → ℙ^{1}, ℤ) = {D ∈ N^{1}(X̄, ℤ) : f_{∗}D = 0} = ℤF + ℤE_{1} + ⋯ + ℤE_{r},

the quotient lattice

N^{1}(X̄ → ℙ^{1}, ℤ)/ℤF = (ℤE_{1} + ℤE_{2} + ⋯ + ℤE_{r} + ℤF)/ℤF

with matrix

\begin{aligned}& E_{1} & E_{2} & … & E_{r} \\ E_{1} & −1 & 0 & … & 0 \\ E_{2} & 0 & −1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ E_{r} & 0 & 0 & … & −1 \end{aligned} ,

and the image Λ of the orthogonal complement K_{X}^{⊥}. This is generated by

ρ_{1} = E′_{1} − E_{2}, ρ_{2} = E_{1} − E_{2}, ρ_{3} = E_{2} − E_{3}, …, ρ_{r} = E_{r−1} − E_{r}

with intersection matrix

ρ_{1} ρ_{2} ρ_{3} ρ_{4} … ρ_{r−1} ρ_{r}
ρ_{1} −2 0 1 0 … 0  0
ρ_{2} 0 −2 1 0 ... ... ⋮
ρ_{3} 1 1 −2 1 ... ... ⋮
ρ_{4} 0 0 1 −2 ... ... 0
⋮  ⋮ ... ... ... ... 1  0
ρ_{r−1} 0 ... ... ... 1 −2 1
ρ_{r} 0 . . . . . . 0 0  1 −2

Up to sign, this is the Cartan matrix associated to the root system D_{r}. Recall the traditional description of D_{r}: Consider

ℤ^{r} = ℤe_{1} + ⋯ + ℤe_{r}

with the standard pairing e_{i} · e_{j} = δ_{ij}. Consider the index two sublattice

M = {m_{1}e_{1} + ⋯ + m_{r}e_{r} : m_{1} + … + m_{r} ≡ 0 (mod 2)} ⊂ ℤ^{r}

with generators

{−e_{1} − e_{2}, e_{1} − e_{2}, e_{2} − e_{3}, …, e_{r−1} − e_{r}}.

The Weyl group W (D_{r}) acts on M via reflections associated to the roots {±e_{i}±e_{j}}. It can be identified with signed r × r permutation matrices with determinant equal to the sign of the permutation. It is thus a semidirect product

W (D_{r}) = (ℤ/2ℤ)^{r−1} ⋊ 𝔖_{r},

where the first group should be interpreted as the diagonal matrices with entries ±1 and determinant 1 and the second group as the permutation matrices. Each element of W (D_{r}) is thus classified by the induced permutation of signed coordinate vectors

{e_{1}, e′_{1} = −e_{1}, …, e_{r}, e′_{r} = −e_{r}}.

Identifying E_{i} with e_{i} and E′_{i} with e′_{i}, we obtain isomorphisms of lattices

M ≃ −Λ
↓  ↓
ℤ^{r} ≃ −N^{1}(f : X̄ → ℙ^{1}, ℤ)/ℤF

where the vertical arrows are inclusions of index-two subgroups. The Galois action of G = Gal(k̄/k) on Pic(X̄) induces actions on both Λ and N^{1}(X̄ → ℙ^{1}, ℤ)/ℤF. It is worthwhile to compare these to the action of W (D_{r}) on M and ℤ^{r}.

Exercises.

Exercise 3.3.1. Let ϕ : X → Y be a birational extremal contraction of smooth projective surfaces, i.e., a contraction of a collection of pairwise disjoint (−1)-curves. Show that

Λ = K_{X}^{⊥} ∩ N^{1}(ϕ : X̄ → Ȳ , ℤ)

is isomorphic to the lattice

ℤρ_{1} + ⋯ + ℤρ_{r−1}

with intersections

ρ_{i} · ρ_{j} = \begin{cases}−2 & if i = j \\ 1 & if |i − j| = 1 \\ 0 & if |i − j| > 1 \end{cases} .

This is the Cartan matrix for A_{r−1}. Interpret the action of the Weyl group W (A_{r−1}) ≃ 𝔖_{r} in terms of the geometry of ϕ.

3.4. Classification of minimal rational surfaces over general fields. This is due to Manin [Man66] and Iskovskikh [Isk79]; another proof can be found in [Kol96, III.2].

Theorem 3.9. Let X be a smooth projective minimal surface with X̄ rational. Then X is one of the following:

• ℙ^{2};

• X ⊂ ℙ^{3} a smooth quadric with Pic(X) = ℤ;

• a Del Pezzo surface with Pic(X) = ℤK_{X};

• a conic bundle f : X → C over a rational curve, with Pic(X) ≃ ℤ ⊕ ℤ.

Notice that the third case includes Brauer-Severi surfaces.

Thus if Y is a smooth projective rational surface over k then there exists a birational morphism ϕ : Y → X defined over k, where X is one of the surfaces listed in Theorem 3.9.

Proof. Since X̄ is rational KX̄ cannot be nef (Exercise 2.4.1), and there exists an irreducible curve L ⊂ X̄ such that KX̄ · L < 0. In particular, \overline{NE}_{1}(X̄) admits KX̄-negative extremal rays. By the Cone Theorem 2.23, elements of \overline{NE}_{1}(X̄) can be expressed as

(3.1)  C + ∑ a_{i}[L_{i}], a_{i} > 0,

where C ∈ \overline{NE}_{1}(X̄) satisfies C · KX̄ ≥ 0 and the L_{i} are rational curves generating KX̄-negative extremal rays. Of course, the Galois group G acts on Pic(X̄) and on the KX̄-negative extremal rays. Thus for elements of \overline{NE}_{1}(X̄) the two parts of (3.1) can be taken to be G-invariant.

Let \overline{NE}_{1}(X̄)^{G} denote the closure of the Galois-invariant effective cone in the real vector space spanned by Galois-invariant curve classes. Since \overline{NE}_{1}(X̄)^{G} has K_{X}-negative curves, it necessarily admits a K_{X}-negative extremal ray Z. This need not be extremal in \overline{NE}_{1}(X̄), but it does lie in some face of that cone, which we analyze. Since Z is extremal and K_{X}-negative, it must be proportional to the average over the orbit of a single extremal ray of X̄:

Z = aE, E = ∑_{j=1}^{n} L_{j}, L_{j} = g_{j}L, g_{j} ∈ G.

In other words, the minimal face of \overline{NE}_{1}(X̄) containing Z is spanned by the Galois orbit of one extremal ray.

Assume first that Pic(X) ≃ ℤ, generated by some ample divisor H defined over k. Then −K_{X} = rH for some positive integer r and X is Del Pezzo. When r > 1 we necessarily have X̄ ≃ ℙ^{2} or ℙ^{1} × ℙ^{1} by Corollary 2.28. In the first instance, X is a Brauer-Severi surface with a line H defined over the ground field; thus Γ(O_{X}(H)) gives an isomorphism X ≃ ℙ^{2}. This is the first case of the theorem. In the second instance, the line bundle O_{ℙ^{1}×ℙ^{1}}(1, 1) is defined over the ground field. Its global sections give an embedding X ↪ ℙ^{3} whose image is a quadric surface. This is the second case of the theorem. Finally, if −K_{X} generates Pic(X) then we are in the third case of the theorem.

Now assume that Pic(X) has higher rank, so in particular E ∈ ∂\overline{NE}_{1}(X̄)^{G}. It follows that E^{2} ≤ 0. Indeed, if E^{2} > 0 then E is big by Corollary 2.4 and thus lies in the interior of the effective cone by Theorem 2.3. (And there are some extremal L whose Galois orbits do not lie in any proper face of the cone of curves.)

Suppose now that E^{2} < 0, which implies that L^{2} < 0. As before, Proposition 1.6 implies L is a (−1)-curve. Furthermore, L ∩ L_{j} = ∅ when L ≠ L_{j}; indeed, if the Galois conjugates were nondisjoint then their sum would have nonnegative self-intersection. Theorem 3.2 implies that X is not minimal, a contradiction.

Suppose next that E^{2} = 0. If L^{2} < 0 then we would still have that L is a (−1)-curve. Since E^{2} = 0 each curve meets precisely one of its Galois conjugates, transversely at one point. Thus the orbit of L decomposes as

{L_{1}, L′_{1}}, {L_{2}, L′_{2}}, …, {L_{r}, L′_{r}},

where L_{i} · L′_{i} = 1 and all other pairs of (−1)-curves are disjoint. Write F_{i} = L_{i} + L′_{i} so that F_{i} · F_{m} = 0 for each i, m = 1, …, r; the Hodge index theorem implies that F_{1} = F_{2} = … = F_{r} and E = rF_{i} for each i. Contracting E (or equivalently, L_{1}, L′_{1}, …, L′_{r}) gives a morphism

f : X → C

whose generic fibers are smooth conics and with r > 0 degenerate fibers consisting of reducible singular conics. This is the conic bundle case of the theorem.

Finally, suppose that E^{2} = 0 and L^{2} = 0. Then each Galois conjugate of L is necessarily disjoint from L, so the Hodge index theorem argument above shows that [L] is Galois-invariant. Contracting L gives a conic bundle f : X → C without degenerate fibers. □

Remark 3.10. This almost completes the birational classification of rational surfaces. It remains to enumerate birational equivalences among the surfaces listed in Theorem 3.9. This enumeration can be found in [MT86, 3.1.1, 3.3.2] Exercises.

Exercise 3.4.1. To get a feeling for the difficulties involved, show that if X is minimal then X̄ ≠ Bl_{p_{1},...,p_{9}}ℙ^{2}. Specify a Galois action on Pic(X̄), in particular, a finite group acting linearly, preserving the intersection form, and fixing K_{X}. Consider the orbits of the (−1)-curves under this action. Convince yourself there is an orbit consisting of either

• disjoint (−1)-curves; or

• r disjoint pairs of (−1)-curves, with each pair meeting transversely at one point.

3.5. An application: Rational points over function fields. Our next result is due to Manin and Colliot-Thélène [CT87]. For more context and discussion, see [Kol96, IV.6]:

Theorem 3.11. Let B be a smooth curve over ℂ with function field k = ℂ(B). Suppose that X is a smooth projective surface over k with X̄ rational. Then X(k) ≠ ∅.

Of course, this result can be obtained from the Graber-Harris-Starr Theorem [GHS03]. However, we will present it using our classification techniques.

Proof. We first reduce to the case where X is minimal. Suppose we have a birational morphism ϕ : X → Y to a smooth projective surface. We can factor ϕ as a sequence

X = X_{0} → X_{1} → X_{2} → ⋯ → X_{r} = Y

where each intermediate morphism is the blow-up of a Galois-orbit of points.

Suppose x ∈ X_{i}(k) is a rational point. If x is contained in the center of the blow-up β_{i} : X_{i−1} → X_{i} then the exceptional divisor E ⊂ X_{i−1} is rational over k and isomorphic to ℙ^{1}. It follows that E(k) ≠ ∅ and X_{i−1}(k) ≠ ∅. If x is disjoint from the center of X_{i−1} → X_{i} then x lies in the open subset U ⊂ X_{i−1} over which β_{i} is an isomorphism. Thus β_{i}^{−1}(x) is a rational point of X_{i}.

We consider the minimal cases one by one. The case X = ℙ^{2} is straightforward. The case of a quadric surface Q ⊂ ℙ^{3} follows from the Tsen-Lang Theorem.

We address the cases of Del Pezzo surfaces of degree d = K_{X}^{2}. Del Pezzo surfaces with certain degrees always have rational points. We assume without proof standard results on anticanonical linear series |−K_{X}| and embeddings of X in projective space:

d = 7 There is no minimal Del Pezzo surface in this degree—see Exercise 3.1.2. d = 8 (X̄ ≃ Bl_{p}ℙ^{2}) There is no minimal Del Pezzo surface of the type—see

Exercise 3.1.3. d = 5 X always has a rational point—see Exercise 3.1.4. d = 1 X̄ ≃ Bl_{p_{1},...,p_{8}}ℙ^{2} in this case and

−KX̄ = 3L − E_{1} − ⋯ − E_{8}.

In this situation, |−K_{X}| is the pencil of cubics C_{t}, t ∈ ℙ^{1}, passing through p_{1}, …, p_{8}. The base locus of this pencil on ℙ^{2} consists of nine points, i.e., p_{1}, …, p_{8} and one additional point p_{0}. The basis locus of |−K_{X}| on X is just the point p_{0}. Since −K_{X} is defined over k, the unique basepoint p_{0} ∈ X(k).

We address the remaining cases using the classification results of §2.6. Since k = ℂ(B), we can use the following variant of the Tsen-Lang Theorem:

Theorem 3.12. Let k be the function field of a curve defined over an algebraically closed field. Let F_{1}, …, F_{r} ∈ k[x_{0}, …, x_{n}] be nonconstant weighted homogeneous polynomials, with weighted degrees satisfying

deg(F_{1}) + ⋯ + deg(F_{r}) ≤ n.

Then the system F_{1} = ⋯ = F_{r} = 0 admits a nontrivial solution over k.

d = 3 Here X is a cubic surface in ℙ^{3} and the result follows from Tsen-Lang. d = 9 X is a Brauer-Severi surface. However, Exercise 3.1.1 allows us to blow

up X to obtain a cubic surface, which has rational points by the previous case. (The reader knowledgeable in central simple algebras can prove Br(ℂ(B)) = 0 using properties of the reduced norm.)

d = 4 Here X is a complete intersection of two quadrics in ℙ^{4} and our variant

of the Tsen-Lang Theorem applies. d = 2 Here |−K_{X}| induces a morphism

X → ℙ^{2}

of degree two, branched over a quartic plane curve. It follows that X is a hypersurface of degree four in the weighted projective space ℙ(2, 1, 1, 1) of the form w^{2} = f(x, y, z). An application of the Tsen-Lang Theorem gives our result.

d = 6 It suffices to show there exists a quadratic extension k′/k over which rational points are dense on X. Then after blowing up two suitable conjugate points we obtain a degree-four Del Pezzo surface, which has a rational point.

From our analysis of the effective cone of X in §2.2, there are two nef divisors L, L′ ∈ Pic(X̄) so that

L^{2} = (L′)^{2} = 1, −K_{X} · L = −K_{X} · L′ = 3.

Indeed, we take L′ = 2L−E_{1} −E_{2} −E_{3}. Their sections induce morphisms

ϕ, ϕ′ : X̄ → ℙ^{2}

blowing down triples of disjoint (−1)-curves. Let k′/k be a quadratic extension over which L and L′ are Gal(k̄′/k′) invariant. Then X_{k′} is a blow-up of a Brauer-Severi variety Y over k′ at three conjugate points. The d = 9 case shows that Y (and hence X_{k′}) has lots of k′-rational points.

For the conic bundle case we apply the Tsen-Lang theorem twice. First, we show that C(k) ≠ ∅ so C ≃ ℙ^{1}. Taking a generic t ∈ ℙ^{1}(k) so that X_{t} := f^{−1}(t) is a smooth conic, a second application gives X_{t}(k) ≠ ∅. □

Exercises.

Exercise 3.5.1. Let X be a degree-one Del Pezzo surface over an arbitrary field k. Give a complete proof that X(k) ≠ ∅, based on the sketch above. Challenge: When can you show that |X(k)| > 1?

4. Singular surfaces

In this section, we work over an algebraically closed field.

Figure 3. Four general lines in the plane

4.1. Cubic surfaces revisited: the Cayley cubic. In §1.1, we constructed smooth cubic surfaces by blowing up six points in general position on the plane. What happens when we relax this assumption?

Consider configurations of six points obtained as pairwise intersections of four general lines in the plane. Given four lines in general position, we can choose coordinates to put them in the standard form:

ℓ_{0} = {x_{0} = 0}, ℓ_{1} = {x_{1} = 0}, ℓ_{2} = {x_{2} = 0}, ℓ_{3} = {x_{0} + x_{1} + x_{2} = 0}.

The intersection points are denoted p_{ij} = ℓ_{i} ∩ ℓ_{j} for 0 ≤ i < j ≤ 3.

The points p_{01}, …, p_{23} still impose independent conditions on homogeneous cubics in x_{0}, …, x_{3}, i.e.,

I_{p_{01},...,p_{23}} = ⟨y_{0}, y_{1}, y_{2}, y_{3}⟩

where

\begin{matrix}y_{0} = x_{1}x_{2}(x_{0} + x_{1} + x_{2}) & y_{1} = x_{0}x_{2}(x_{0} + x_{1} + x_{2}) \\ y_{2} = x_{0}x_{1}(x_{0} + x_{1} + x_{2}) & y_{3} = −x_{0}x_{1}x_{2} \end{matrix} .

These satisfy the relation

y_{0}y_{1}y_{2} + y_{1}y_{2}y_{3} + y_{2}y_{3}y_{0} + y_{3}y_{0}y_{1} = 0;

the resulting cubic surface S ⊂ ℙ^{3} is called the Cayley cubic surface in honor of Arthur Cayley, who classified singular cubic surfaces [Cay69].

Here are some of its geometric properties: • S has ordinary double points at

s_{0} = [1, 0, 0, 0], s_{1} = [0, 1, 0, 0], s_{2} = [0, 0, 1, 0], s_{3} = [0, 0, 0, 1].

Figure 4. Some lines on the Cayley cubic

It is the unique cubic surface with this configuration of singularities, up to projective equivalence.

• S contains nine lines, i.e., the lines m_{ij}, i, j = 0, …, 3 spanned by s_{i} and s_{j}, as well as the lines

{y_{0} + y_{3} = y_{1} + y_{2} = 0}, {y_{0} + y_{1} = y_{2} + y_{3} = 0}, {y_{0} + y_{2} = y_{1} + y_{3} = 0}.

• The birational map

[y_{0}, y_{1}, y_{2}, y_{3}] : ℙ^{2} −−→^{∼} S

factors as

X
↙^{β}  ↘^{σ}
ℙ^{2}  S

where β is the blow-up of p_{01}, …, p_{23} and σ is the blow-up of s_{0}, …, s_{3}. The exceptional divisors of β are the proper transforms E_{ij} of the m_{ij}; the exceptional divisors of σ are the proper transforms ℓ′_{i} of the ℓ_{i}.

• Express

Pic(X) = ℤL ⊕ ℤE_{01} ⊕ ⋯ ⊕ ℤE_{23}

where L is the pullback of the hyperplane class of ℙ^{2} via β. The canonical class is

K_{X} = −3L + E_{01} + E_{02} + E_{03} + E_{12} + E_{13} + E_{23}

and the proper transforms of the lines are

ℓ′_{0} = L − E_{01} − E_{02} − E_{03}, ℓ′_{1} = L − E_{01} − E_{12} − E_{13}, …

We have K_{X} · ℓ′_{j} = 0 and (ℓ′_{j})^{2} = −2 for each j, i.e., the exceptional divisors of the resolution σ are (−2)-curves.

4.2. Why consider singular cubic surfaces? Reason 1: Reduction modulo primes Let X = {F (y_{0}, y_{1}, y_{2}, y_{3}) = 0} ⊂ ℙ^{3} be a smooth cubic surface defined over ℚ; we may assume that F ∈ ℤ[y_{0}, y_{1}, y_{2}, y_{3}] and the greatest common divisor of the coefficients of F is 1. Consider the integral model

π : X = {F = 0} ⊂ ℙ_{ℤ}^{3} → Spec(ℤ),

which is flat and projective over Spec(ℤ). For each prime p, we have X_{p} = X (mod p), i.e., the fiber of X over p ∈ Spec(ℤ). If p divides the discriminant of F then X_{p} will have singularities. These singular fibers have a strong influence on the rational points of X. Reason 1′: Degenerate fibers of families This is the function-field analog of the previous situation. Let B be a complex curve and

π : X → B

a family of cubic surfaces, e.g., a pencil

{sF (y_{0}, y_{1}, y_{2}, y_{3}) + tG(y_{0}, y_{1}, y_{2}, y_{3}) = 0} ⊂ ℙ_{y_{0},y_{1},y_{2},y_{3}}^{3} × ℙ_{s,t}^{1}

with π being projection onto the second factor. At least some of the fibers X_{b} = π^{−1}(b), b ∈ B must be singular. Reason 2: Counting rational points Proving asymptotics for the number of rational points of bounded heights on singular cubic surfaces is often easier than the case of smooth cubic surfaces. Examples include toric cubic surfaces [dlB98, Fou98, HBM99, Sal98]

y_{0}^{3} = y_{1}y_{2}y_{3},

the Cayley cubic surface [HB03], the ‘E_{6} cubic surface’ [dlBBD07]

y_{1}y_{2}^{2} + y_{2}y_{0}^{2} + y_{3}^{3} = 0;

and a ‘D_{4} cubic surface’ [Bro06]

y_{1}y_{2}y_{3} = y_{4}(y_{1} + y_{2} + y_{3})^{2}.

4.3. What are ‘good’ singularities? Let S be a normal surface. A resolution of singularities σ : X → S is a birational proper morphism from a smooth surface. Abhyankar proved the existence of resolutions of surface singularities in arbitrary characteristic [Abh56]. A resolution σ : X → S is minimal if there exists no nontrivial factorization

X →^{ϕ} Y → S

with Y smooth. This is equivalent to • there are no (−1)-curves in the fibers of σ; or

• K_{X} is nef relative to σ. A relative analog of Corollary 2.20 (see Remark 2.21) implies that minimal resolutions of singularities are unique, in the case of surfaces.

Recall that if ϕ : X → Y is a birational morphism of smooth projective surfaces then (cf. Equation 1.1):

K_{X} = ϕ^{∗}K_{Y} + ∑_{i} m_{i}E_{i}, m_{i} > 0.

The following definition represents a weakening of this condition:

Definition 4.1. Suppose that S is a normal surface with a unique singularity p; assume that K_{S} is a ℚ-Cartier divisor at p. (This is the case when S is a complete intersection in some neighborhood of p). Then p ∈ S is a canonical singularity if, for each resolution of singularities σ : X → S we have

K_{X} = σ^{∗}K_{S} + ∑_{i} m_{i}E_{i}, m_{i} ≥ 0,

where the E_{i} are exceptional divisors of σ.

Note here that a priori the m_{i} ∈ ℚ; however, the classification of these singularities shows a posteriori that the m_{i} ∈ ℤ.

Proposition 4.2. Suppose that (S, p) is a canonical singularity. A resolution of singularities σ : X → S is minimal if and only if each m_{i} = 0, i.e., K_{X} = σ^{∗}K_{S}. In this case, each σ-exceptional curve is a (−2)-curve, i.e., a nonsingular rational curve E with E^{2} = −2.

Proof. (⇐) Suppose that m_{i} = 0 for each i. Then K_{X} · E_{i} = 0 for each σ-exceptional divisor. The Hodge index theorem implies that the σ-exceptional divisors have negative self-intersection, i.e., E_{i}^{2} < 0. The adjunction formula implies that E_{i}^{2} = −2 and E_{i} is a nonsingular curve of arithmetic genus zero.

(⇒) Assume that σ is minimal, i.e., the fibers of σ contain no (−1)-curves. Suppose that K_{X} ≠ σ^{∗}K_{S} so that some m_{i} ≠ 0. It follows that (∑_{i} m_{i}E_{i})^{2} < 0 and thus (∑_{i} m_{i}E_{i})·E_{j} < 0 for some σ-exceptional curve E_{j}. Consequently E_{j}^{2} < 0 and K_{X}E_{j} < 0, so E_{j} is a (−1)-curve by Proposition 1.6. □

Proposition 4.2 suggests the following variation on this definition

Definition 4.3. A normal surface S has Du Val singularities if it admits a resolution σ : X → S such that K_{X} · E = 0 for each σ-exceptional divisor E.

Patrick Du Val first classified surface singularities in terms of their discrepancies (or in his terminology, the ‘conditions they impose on adjunction’) in [DV34]. This definition is a priori more general than the class of canonical singularities: We do not insist that K_{S} is ℚ-Cartier. However, we shall see later (Remark 4.7) that Du Val singularities are canonical.

Exercises.

Exercise 4.3.1. We give an example of a surface with ‘bad’ singularities. Suppose that p_{1}, …, p_{4} ∈ ℓ ⊂ ℙ^{2} are distinct points lying on a line ℓ. Consider

β : X := Bl_{p_{1},...,p_{4}}ℙ^{2} → ℙ^{2}

and let ℓ̃ denote the proper transform of ℓ, L the pullback of the hyperplane class via β, and E_{1}, …, E_{4} the exceptional divisors. Verify that

a. The divisor 4L−E_{1}−E_{2}−E_{3}−E_{4} is basepoint-free and yields a morphism

ϕ : X → ℙ^{10}. b. If Y is the image of X under ϕ, show that ϕ : X → Y is an isomorphism

over X \ ℓ̃ and contracts ℓ̃ to a point y ∈ Y. c. Show that Y is normal at y and the canonical class K_{Y} is ℚ-Cartier.

Compute the divisor ϕ^{∗}K_{Y}. e. Show that y ∈ Y is not a Du Val singularity.

4.4. Singular Del Pezzo surfaces.

Definition 4.4. A singular Del Pezzo surface is a projective surface S with Du Val singularities such that −K_{S} is ample.

If σ : X → S is a minimal resolution of a singular Del Pezzo surface then σ^{∗}K_{S} = K_{X}, i.e., −K_{X} is semiample.

Here is one good source of singular Del Pezzo surfaces. Suppose X is a smooth projective surface with −K_{X} nef and big. It has the following properties:

• (−K_{X})^{2} > 0;

• Any irreducible curve E with K_{X} · E = 0 is a (−2)-curve.

• There are a finite number of (−2)-curves on X. The first statement is a particular case of Corollary 2.4. The second is contained in the proof of Proposition 4.2. The third follows from the fact that K_{X}^{⊥} is negative definite, and thus has a finite number of vectors of self-intersection −2.

Theorem 4.5. Let X be a smooth projective surface with −K_{X} nef and big. Then each nef divisor D on X is semiample.

Corollary 4.6. Let X be a smooth projective surface with −K_{X} nef and big. Then −K_{X} is semiample. In particular, there exists a birational morphism σ : X → S to a singular Del Pezzo surface with σ^{∗}K_{S} = K_{X}.

Proof. Remark 2.18 addresses this in the special case where D is not big, i.e., when D^{2} = 0. Thus we may assume that D^{2} > 0.

The Nakai criterion (Theorem 2.2) implies that D is ample unless D · E = 0 for some irreducible curve E ⊂ X. The Hodge index theorem implies that each such curve satisfies E^{2} < 0. Since −K_{X} · E ≥ 0, the only possibilities are (−1)-curves (see Proposition 1.6) or (−2)-curves (see Proposition 4.2). In either case E ≃ ℙ^{1}.

Suppose X admits (−1)-curves as above. We can apply the Castelnuovo contraction criterion (Theorem 1.12) to obtain a birational morphism β : X → Y such that Y admits a big and nef divisor M on Y with β^{∗}M = D and the only curves orthogonal to M are (−2)-curves. Furthermore, −K_{Y} remains nef and big (cf. Corollary 2.8).

Let E_{1}, …, E_{r} denote the (−2)-curves orthogonal to M. We exhibit a birational morphism to a singular projective variety σ : Y → S contracting precisely these curves. Such a contraction exists for more general reasons [Rei97, 4.15] [Art62, 2.3] but we will sketch an argument in our situation.

We essentially copy the proof of the Castelnuovo Criterion. Let H be a very ample line bundle on Y such that each positive multiple nH has no higher cohomology. Write d_{i} = H · E_{i} for i = 1, …, r. Since the intersection matrix of ℤE_{1} + ⋯ + ℤE_{r} is negative definite, there exist positive integers n and b_{1}, …, b_{r} such that

nH · E_{i} = −(b_{1}E_{1} + ⋯ + b_{r}E_{r}) · E_{i}

for each i. Let B = b_{1}E_{1} + ⋯ + b_{r}E_{r} so that L := nH + B is orthogonal to each E_{i}.

The adjunction formula implies that each effective divisor A supported on E_{1} ∪ ⋯ ∪ E_{r} has nonpositive arithmetic genus; a straightforward induction gives H^{1}(O_{A}) = 0 as well. Here it is crucial that K_{Y} · E_{i} = 0 for each i; it is not enough to assume that each component of the exceptional locus is rational. Thus we have O_{Y} (L)|B ≃ O_{B}, i.e., an isomorphism of invertible sheaves, not just an equality of degrees. We obtain the exact sequence

0 → O_{Y} (nH) → O_{Y} (L) → O_{B} → 0.

Our vanishing assumption show that

Γ(Y, O_{Y} (L)) ↠ Γ(Y, O_{B}),

i.e., for each point of B there is a section of O_{Y} (L) nonvanishing at that point. The sections of O_{Y} (L) induce an embedding away from E_{1} ∪ ⋯ ∪ E_{r}, so O_{Y} (L) is globally generated and induces a morphism σ : Y → S contracting precisely E_{1}, …, E_{r}. In particular, O_{Y} (L) is the pullback of an ample line bundle on S via σ.

To complete the argument, we show there exists a Cartier divisor N on S such that σ^{∗}N is a positive multiple of M. Repeating the previous argument for M +mL with m ≫ 0, we get the same contraction σ : Y → S. Here the argument shows that mL + M is the pullback of an ample line bundle from S. It follows that M = σ^{∗}N for some Cartier divisor N on X.

Finally, N is ample on S by the Nakai criterion, as we have contracted all the curves along which it is nonpositive. □

Remark 4.7. A variation on this argument shows that Du Val singularities are canonical. Suppose that σ : Y → S is a minimal resolution of Du Val singularities. The canonical class K_{Y} is nef relative to σ and thus globally generated relative to σ. We obtain a factorization

Y → Proj_{S} (⨁_{n≥0} σ_{∗}O_{Y} (nK_{Y})) →^{ϖ} S.

Since ϖ is a bijective morphism of normal surfaces, it is an isomorphism. However, the canonical divisor of the intermediate surface is ℚ-Cartier by construction.

Remark 4.8. Suppose the base field is algebraically closed of characteristic zero. There do exist smooth projective rational surfaces admitting nef divisors that are not semiample [Zar62, §2]. Thus the assumption that −K_{X} be nef and big in Theorem 4.5 is necessary. (See Exercise 4.4.1 below and [Laz04, 2.3] for more discussion.)

We record one last consequence of Theorem 4.5, an extension of Corollary 2.13:

Proposition 4.9. Let X be a smooth projective surface with −K_{X} nef and big. Then \overline{NE}_{1}(X) is a finite rational polyhedral cone, generated by (−2)-curves and K_{X}-negative extremal rational curves.

Proof. Apply Proposition 2.10 and Corollary 2.11: \overline{NE}_{1}(X) is generated by the nonnegative cone \overline{C}, along with the (−1)-curves and (−2)-curves. The Hodge index theorem implies that K_{X} is negative on \overline{C} \ \{0\}, so any extremal rays of \overline{NE}_{1}(X) arising from \overline{C} are necessarily K_{X}-negative.

The Cone Theorem 2.23 implies that the K_{X}-negative part of the effective cone is generated by curves L_{i} with −K_{X} · L_{i} ≤ 3. Theorem 4.5 gives that −K_{X} is semiample and induces σ : X → S. Thus there are at most a finite number of classes [L_{i}] arising as K_{X}-negative extremal rays. Indeed, the curves in S with anticanonical degree ≤ 3 are parametrized by a scheme of finite type, as the curves in projective space of bounded degree are parameterized by a Hilbert scheme of finite type. We see in particular that X admits a finite number of (−1)-curves.

Clearly, there are a finite number of (−2)-curves, as these are all σ-exceptional. Thus \overline{NE}_{1}(X) admits a finite number of extremal rays, with the desired interpretations. □

Exercises.

Exercise 4.4.1. Assume the base field is of characteristic zero.

Let C ⊂ ℙ^{2} denote a smooth cubic plane curve and H the hyperplane class on ℙ^{2}. Choose points p_{1}, …, p_{9} ∈ C such that the divisors p_{1} + ⋯ + p_{9} and H|C are linearly independent over ℚ. Consider the blow-up

X := Bl_{p_{1},...,p_{9}}ℙ^{2} →^{β} ℙ^{2}

with exceptional curves E_{1}, …, E_{9}. Show that D = −K_{X} = 3β^{∗}H − E_{1} − ⋯ − E_{9} is nef but not semiample.

Now choose points q_{1}, …, q_{12} ∈ C such that q_{1} + ⋯ + q_{12} and H|C are linearly independent. Consider the blow-up

Y := Bl_{q_{1},...,q_{12}}ℙ^{2} →^{γ} ℙ^{2}

with exceptional curves F_{1}, …, F_{12}. Show that D′ = 4γ^{∗}H − F_{1} − ⋯ − F_{12} is nef but not semiample. Indeed, demonstrate that for each n > 0 the divisor nD′ has the proper transform of C as a fixed component.

4.5. Classification of Du Val singularities. Suppose that σ : X → S is a minimal resolution of a Du Val surface singularity p ∈ S. Consider the intersection numbers of the irreducible components E_{1}, …, E_{r} of σ^{−1}(p), which we put into a symmetric matrix (E_{i} · E_{j})_{i,j=1,...,r}. This has the following properties:

• (E_{i} · E_{j}) is negative definite, by the Hodge index theorem;

• E_{i}^{2} = −2 for each i, by Proposition 4.2;

• E_{i} · E_{j} = 0, 1 for each i ≠ j; indeed, if E_{i} · E_{j} > 1 then (E_{i} + E_{j})^{2} > 0;

• we cannot express

{E_{1}, …, E_{r}} = {E_{a_{1}}, …, E_{a_{s}}} ∪ {E_{b_{1}}, …, E_{b_{r−s}}}

with E_{a_{l}} · E_{b_{m}} = 0 for each l, m; this is because σ^{−1}(p) is connected. Matrices of this type occur throughout mathematics, especially in the classi-

fication of the simple root systems via Dynkin diagrams/Cartan matrices in Lie theory. We cannot dwell too much on these interactions, except to refer the reader to some of the literature on this beautiful theory [Bri71, Dur79, SB01]. We list the possible matrices that can arise [FH91, 21.2]. First, we have the infinite series

\begin{matrix}A_{r} \\ r ≥ 1 \end{matrix} (\begin{matrix}−2 & 1 & 0 & … & … & 0 \\ 1 & −2 & 1 & ... & ... & ⋮ \\ 0 & 1 & −2 & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & 1 & 0 \\ ⋮ & ... & ... & 1 & −2 & 1 \\ 0 & … & … & 0 & 1 & −2 \end{matrix}) E_{i} · E_{i} = \begin{cases}−2 & if i = j \\ 1 & if |i − j| = 1 \\ 0 & otherwise \end{cases}

\begin{matrix}D_{r} \\ r ≥ 4 \end{matrix} (\begin{matrix}−2 & 0 & 1 & 0 & … & 0 & 0 \\ 0 & −2 & 1 & 0 & ... & ... & ⋮ \\ 1 & 1 & −2 & 1 & ... & ... & ⋮ \\ 0 & 0 & 1 & −2 & ... & ... & 0 \\ ⋮ & ... & ... & ... & ... & 1 & 0 \\ 0 & ... & ... & ... & 1 & −2 & 1 \\ 0 & … & … & 0 & 0 & 1 & −2 \end{matrix}) E_{i} · E_{i} = \begin{cases}−2 & if i = j \\ 1 & if |i − j| = 1, \\ & i, j ≥ 3 \\ & or if \{i, j\} \\ & = \{1, 3\}, \{2, 3\} \\ 0 & otherwise \end{cases}

and then the exceptional lattices

E_{6} (\begin{matrix}−2 & 1 & 0 & 0 & 0 & 0 \\ 1 & −2 & 0 & 0 & 0 & 1 \\ 0 & 0 & −2 & 1 & 0 & 0 \\ 0 & 0 & 1 & −2 & 0 & 1 \\ 0 & 0 & 0 & 0 & −2 & 1 \\ 0 & 1 & 0 & 1 & 1 & −2 \end{matrix})

E_{7} (\begin{matrix}−2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & −2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & −2 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & −2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & −2 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & −2 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & −2 \end{matrix})

E_{8} (\begin{matrix}−2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & −2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & −2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & −2 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & −2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & −2 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & −2 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & −2 \end{matrix})

Remarkably, in characteristic zero there is a unique singularity associated to each of these matrices.

Proposition 4.10. Assume that the base field is algebraically closed of characteristic zero. Then, up to analytic isomorphism, there is a unique Du Val surface singularity associated to each Cartan matrix enumerated above:

\begin{matrix}A_{r}, r ≥ 1 & z^{2} = x^{2} + y^{r+1} \\ D_{r}, r ≥ 4 & z^{2} = y(x^{2} + y^{r−2}) \\ E_{6} & z^{2} = x^{3} + y^{4} \\ E_{7} & z^{2} = y(x^{3} + y^{2}) \\ E_{8} & z^{2} = x^{3} + y^{5} \end{matrix}

For a modern proof of this, we refer the reader to [KM98, §4.2]. It turns out that these singularities are also related to the class of ‘simple’ hypersurface singularities, which can be independently classified [AGZV85]. There are a multitude of classical characterizations of Du Val singularities [Dur79].

Example 4.11. The Cayley cubic surface has four A_{1} singularities. The toric cubic surface

y_{0}^{3} = y_{1}y_{2}y_{3}

has three A_{2} singularities at [0, 1, 0, 0],[0, 0, 1, 0], and [0, 0, 0, 1].

5. Cox rings and universal torsors

We work over an algebraically closed field k unless specified otherwise.

5.1. Universal torsors. Universal torsors are an important tool in higherdimensional arithmetic geometry. They play a fundamental rôle in the modern theory of descent for rational varieties [CTS87]. They are also an important technique and conceptual tool for counting rational points of bounded height [Sal98].

Let X be a smooth projective variety. Assume that Pic(X) is a free abelian group of rank r, generated by the line bundles L_{1}, …, L_{r} on X. Let T_{X} = Hom(Pic(X), 𝔾_{m}) denote the Néron-Severi torus of X, i.e., the torus with character group Hom(T_{X}, 𝔾_{m}) = Pic(X).

Definition 5.1. [CTS87] The universal torsor over X

\begin{aligned}T_{X} & → & U \\ & & ↓ \\ & & X \end{aligned}

is a principal homogeneous space over X with structure group T_{X} with the following universal property: Given a line bundle L on X, if λ_{L} : T_{X} → 𝔾_{m} = GL_{1} denotes the corresponding character then the line bundle V_{λ_{L}} associated to U equals L. In other words, if U is given by a cocycle {τ_{ij}} ∈ H^{1}(X, T_{X}) then L is given by the cocycle {λ_{L}(τ_{ij})} ∈ H^{1}(X, 𝔾_{m}).

Constructing U is straightforward in some sense: Choose L_{1}, …, L_{r} freely generating Pic(X) and write

P_{i} = L_{i}^{−1} \ 0_{X} ⊂ L_{i}^{−1}

for the complement of the zero-section. This is a 𝔾_{m}-principal bundle arising from L_{i}^{−1}. Then we can take

U = P_{1} ×_{X} ⋯ ×_{X} P_{r}

and T_{X}-action

\begin{matrix}T_{X} × U & → & U \\ (t; s_{1}, …, s_{r}) & ↦ & (λ_{−L_{1}}(t)s_{1}, …, λ_{−L_{r}}(t)s_{r}) \end{matrix}

where s_{i} is a local section of P_{i} and λ_{L} is the character associated with L.

However, for arithmetic applications it is important to have a more concrete presentation of the universal torsor.

Example 5.2. Consider the case X = ℙ^{n}. The standard quotient presentation

ℙ^{n} = (𝔸^{n+1} \ 0)/𝔾_{m}

can be interpreted as an identification:

O_{ℙ^{n}}(−1) \ 0_{ℙ^{n}}  →^{∼}  𝔸^{n+1} \ 0
↘  ↙
ℙ^{n}

In other words, we regard O_{ℙ^{n}}(−1) as the ‘universal line’ over ℙ^{n}. Since O_{ℙ^{n}}(−1) generates Pic(ℙ^{n}), we have

U = O_{ℙ^{n}}(−1) \ 0_{ℙ^{n}} = 𝔸^{n+1} \ 0,

equivariant with respect to the action of 𝔾_{m} = T_{ℙ^{n}}. Note that we can regard

𝔸^{n+1} = Spec (⨁_{N∈ℤ} Γ(ℙ^{n}, O_{ℙ^{n}}(N))) .

More generally, the universal torsor

T_{ℙ^{m}×ℙ^{n}} →  U
↓
ℙ^{m} × ℙ^{n}

can be identified with

𝔸_{x_{0},...,x_{m},y_{0},...,y_{n}}^{m+n+2} \ ({x_{0} = ⋯ = x_{m} = 0} ∪ {y_{0} = ⋯ = y_{n} = 0}) .

Here the torus acts by the rule

(t_{1}, t_{2}) · (x_{0}, …, x_{m}, y_{0}, …, y_{n}) = (t_{1}x_{0}, …, t_{1}x_{m}, t_{2}y_{0}, …, t_{2}y_{n}).

Decomposing the polynomial ring under this action, we can regard

𝔸^{m+n+2} = Spec ( ⨁_{N_{1},N_{2}∈ℤ} Γ(ℙ^{m} × ℙ^{n}, O_{ℙ^{m}×ℙ^{n}}(N_{1}, N_{2}))) .

Exercises.

Exercise 5.1.1. Realize the universal torsor over X = ℙ^{1} × ℙ^{1} × ℙ^{1} as an explicit open subset U ⊂ 𝔸^{8}. Describe the action of T_{X} on U.

5.2. Universal torsors over nonclosed fields. We can only offer a brief summary here; we refer the reader to [CTS87] and [Sko01] for details and arithmetic applications.

Let k be a perfect field with absolute Galois group G. Suppose that X is defined over k and X̄ satisfies the assumptions made in §5.1. The Galois action on Pic(X̄) allows us to define the torus T_{X} over k. Precisely, the action induces a representation on the character group

ϱ : G → Aut(Hom(TX̄, 𝔾_{m})) = Aut(Pic(X̄)),

which gives the descent data for T_{X}. A universal torsor over X is a principal homogeneous space

\begin{aligned}T_{X} & → & U \\ & & ↓ \\ & & X \end{aligned}

defined over k, such that the universal property is satisfied on passage to the algebraic closure.

Note our use of the indefinite article: Over a nonclosed field, a variety may have more than one universal torsor. Indeed, since any universal torsor U comes with a T_{X}-action over X, given a cocycle η ∈ H_{G}^{1}(TX̄) we can twist to obtain

\begin{aligned}T_{X} & → & U^{η} \\ & & ↓ \\ & & X, \end{aligned}

another universal torsor over k. However, if the Galois action on Pic(X̄) is trivial then

H_{G}^{1}(TX̄) = H_{G}^{1}(𝔾_{m}^{r}) = 0

by Hilbert’s Theorem 90. Here the universal torsor is unique whenever it exists.

On the other hand, there may be obstructions to descending a universal torsor over X̄ to the field k. These reside in H_{G}^{2}(TX̄); indeed, the situation is analogous to the descent obstruction for line bundles discussed in Remark 3.3. Whenever X(k) ≠ ∅ this obstruction vanishes [CTS87, 2.2.8], which makes universal torsors an important tool for deciding whether X has rational points.

5.3. Cox rings. Let X be a normal projective variety such that the Weil divisor class group is freely generated by D_{1}, …, D_{r}.

Definition 5.3. The Cox ring of X is defined as

Cox(X) = ⨁_{(n_{1},...,n_{r})∈ℤ^{r}} Γ(X, O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r}))

with multiplication

\begin{matrix}Γ(X, O_{X}(m_{1}D_{1} + ⋯ + m_{r}D_{r})) × Γ(X, O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})) & → \\ Γ(X, O_{X}((m_{1} + n_{1})D_{1} + ⋯ + (m_{r} + n_{r})D_{r})) \end{matrix}

defined by (s, t) ↦ st.

Example 5.4. We start with the eponymous example [Cox95]: Let X be a projective toric variety of dimension n

𝔾_{m}^{n} × X → X.

Let D_{1}, …, D_{d} denote the boundary divisors, i.e., the irreducible components of the complement of the dense open torus orbit. Let s_{i} ∈ Γ(X, O_{X}(D_{i})) denote the canonical section, i.e., the one associated with the inclusion

O_{X} ↪ O_{X}(D_{i}).

(Actually, s_{i} is canonical up to a nonzero scalar.) Recall that

• each effective divisor D on X can be expressed as a nonnegative linear combination

D ≡ n_{1}D_{1} + ⋯ + n_{d}D_{d}, n_{1}, …, n_{d} ≥ 0;

• the canonical section s of O_{X}(D) admits a unique expression

s = f(s_{1}, …, s_{d})

where f is a polynomial over k in d variables.

Thus the Cox ring of X is a polynomial ring

Cox(X) ≃ k[s_{1}, …, s_{d}]

with generators indexed by the boundary divisors.

We list some basic properties of the Cox ring of a smooth projective variety. We continue to assume that D_{1}, …, D_{r} are divisors freely generating Pic(X).

• Cox(X) is graded by Pic(X), i.e.,

Cox(X) ≃ ⨁_{L∈Pic(X)} Cox(X)_{L}, Cox(X)_{L} ≃ Γ(X, L).

Indeed, for a unique choice of (n_{1}, …, n_{r}) ∈ ℤ^{r} we have an isomorphism

L ≃ O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r}).

• Cox(X) has a natural action by T_{X} via the rule

(t_{1}, …, t_{r}) · s = t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}}s

when s ∈ Γ(X, O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})).

• The nonzero graded pieces of Cox(X) are indexed by NE^{1}(X, ℤ). If Cox(X) is finitely generated then \overline{NE}^{1}(X) is a finitely generated rational polyhedral cone.

5.4. Two theorems. We start with a general result:

Proposition 5.5. Let X be a projective variety and A_{1}, …, A_{r} semiample Cartier divisors on X. Then the ring

(5.1)  ⨁_{n_{1},...,n_{r}≥0} Γ(O_{X}(n_{1}A_{1} + ⋯ + n_{r}A_{r}))

is finitely generated.

Proof. (based on [HK00, 2.8], with suggestions from A. Várilly-Alvarado) It suffices to show that for some positive N ∈ ℕ the ring

⨁_{n_{1},...,n_{r}≥0} Γ(O_{X}(N(n_{1}A_{1} + ⋯ + n_{r}A_{r})))

is finitely generated. Indeed, the full ring is integral over this subring, so our result follows by finiteness of integral closure. Since A_{1}, …, A_{r} are semiample, there exists an N > 0 such that NA_{1}, …, NA_{r} are globally generated. Thus we may assume that A_{1}, …, A_{r} are globally generated.

We first consider the special case r = 1. We obtain a morphism

ϕ : X → ℙ^{m} := ℙ(Γ(O_{X}(A_{1}))^{∗}),

with ϕ^{∗}O_{ℙ^{m}}(1) = O_{X}(A_{1}). This admits a Stein factorization

X →^{f} Y →^{g} ℙ^{m}

with g finite and f having connected fibers, so in particular f_{∗}O_{X} = O_{Y}. Furthermore, g^{∗}O_{ℙ^{m}}(1) is ample on Y and thus

⨁_{n≥0} Γ(Y, g^{∗}O_{ℙ^{m}}(n))

is finitely generated. The projection formula gives

g^{∗}Γ(Y, g^{∗}O_{ℙ^{m}}(n)) →^{∼} Γ(X, O_{X}(nA_{1}))

for each n ∈ ℕ, so

⨁_{n≥0} Γ(X, O_{X}(nA_{1}))

is also finitely generated.

Now suppose r is arbitrary. Consider the vector bundle

V = A_{1} ⊕ ⋯ ⊕ A_{r}

and the associated projective bundle

π : ℙ(V^{∗}) → X.

We have the tautological quotient bundle

π^{∗}V ↠ O_{ℙ(V^{∗})}(1);

since V is globally generated (being the direct sum of globally generated line bundles), O_{ℙ(V^{∗})}(1) is semiample. In particular, the ring

(5.2)  ⨁_{n≥0} Γ(ℙ(V^{∗}), O_{ℙ(V^{∗})}(n))

is finitely generated.

The tautological quotient induces

Sym^{n}π^{∗}V ↠ O_{ℙ(V^{∗})}(n),

and taking direct images via the projection formula we obtain

Sym^{n}V →^{∼} π_{∗}O_{ℙ(V^{∗})}(n)

and

Γ(X, Sym^{n}V) = Γ(ℙ(V^{∗}), O_{ℙ(V^{∗})}(n)).

Since (5.2) is finitely generated, the algebra

⨁_{n≥0} Γ(X, Sym^{n}V)

is finitely generated as well. Using the decomposition

Sym^{n}V = ⨁_{n_{1}+···+n_{r}=n} ; n_{1},...,n_{r}≥0 O_{X}(n_{1}A_{1} + ⋯ + n_{r}A_{r}),

we conclude that (5.1) is finitely generated. □

Remark 5.6 (due to A. Várilly-Alvarado). If A_{1} and A_{2} are ample then there exists an N ∈ ℕ such that the multiplication maps

Γ(X, O_{X}(Nm_{1}A_{1})) ⊗ Γ(X, O_{X}(Nm_{2}A_{2})) → Γ(X, O_{X}(N(m_{1}A_{1} + m_{2}A_{2})))

are surjective for each m_{1}, m_{2} ≥ 0. However, this fails for semiample divisors.

Let h : X → ℙ^{1}×ℙ^{1} be a double cover branched over a smooth curve of bidegree (2d, 2d); composing with the projections yield morphisms g_{i} : X → ℙ^{1}, i = 1, 2, with connected fibers. Let f_{1} and f_{2} be the fibers of ℙ^{1} × ℙ^{1}; take A_{1} and A_{2} to be their preimages on X. Then we have

Γ(X, O_{X}(mA_{1})) = Γ(ℙ^{1}, O_{ℙ^{1}}(m))

for each m ≥ 0, i.e., sections of

Γ(X, O_{X}(m_{1}A_{1})) ⊗ Γ(X, O_{X}(m_{2}A_{2}))

are obtained via pullback from sections of Γ(ℙ^{1} × ℙ^{1}, O_{ℙ^{1}×ℙ^{1}}(m_{1}, m_{2})). Since m_{1}A_{1} + m_{2}A_{2} is very ample on X for suitable m_{1}, m_{2} ≫ 0, we conclude that

Γ(X, O_{X}(m_{1}A_{1})) ⊗ Γ(X, O_{X}(m_{2}A_{2})) → Γ(X, O_{X}(m_{1}A_{1} + m_{2}A_{2}))

cannot be surjective. Indeed, the decomposable sections cannot separate points in a fiber of h.

Theorem 5.7. Suppose X is a smooth projective variety with Pic(X) free of finite rank. Assume that Cox(X) is finitely generated. Then the universal torsor admits an embedding

ι : U ↪ Spec(Cox(X))

that is equivariant under the action of the Néron-Severi torus T_{X}.

Proof. First, we construct the morphism ι. Again, let D_{1}, …, D_{r} denote divisors freely generating the divisor class group of X. The cone of effective divisors of X is finite rational polyhedral and strictly convex, so we can choose D_{1}, …, D_{r} such that each effective divisor on X can be written as a nonnegative linear combination of D_{1}, …, D_{r}. (Of course, the D_{i} themselves need not be effective.)

Let L_{1}, …, L_{r} designate the line bundles associated to the invertible sheaves O_{X}(D_{1}), …, O_{X}(D_{r}). Writing P_{i} = L_{i}^{−1} \ 0_{X} we have

U = P_{1} ×_{X} ⋯ ×_{X} P_{r} ⊂ L_{1}^{−1} ×_{X} ⋯ ×_{X} L_{r}^{−1}

which we interpret as the natural inclusion of

Spec_{X} ( ⨁_{(n_{1},...,n_{r})∈ℤ^{r}} O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r}))

into

Spec_{X} ( ⨁_{n_{1},...,n_{r}≥0} O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})) .

For each (n_{1}, …, n_{r}) ∈ ℤ_{≥0}^{r}, we have

Γ(X, O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})) ⊗ O_{X} → O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})

which induces

Spec_{X} (⨁_{(n_{1},...,n_{r})∈ℤ_{≥0}^{r}} O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})) ⟶
Spec_{X} (⨁_{(n_{1},...,n_{r})∈ℤ_{≥0}^{r}} Γ(O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})) ⊗ O_{X}) .

Since each effective divisor is a nonnegative sum of the D_{i}, the target is isomorphic to X × Spec(Cox(X)). Thus we get a morphism

U  →  X × Spec(Cox(X))
↘  ↙
X

and composing with the projection yields

ι : U → Spec(Cox(X)).

Our construction is clearly equivariant with respect to the actions of T_{X}.

We prove ι is an open embedding. First, observe that Spec(Cox(X)) is normal, i.e., Cox(X) is integrally closed in its fraction field. Since X is normal,

⨁_{n_{1},...,n_{r}≥0} O_{X}(n_{1}D_{1} + ⋯ + n_{r}D_{r})

is a sheaf of integrally-closed domains, whose global sections form an integrally closed domain (cf, [Har77, Ex. 5.14(a)]). Furthermore, Cox(X) is even a UFD [EKW04, Cor. 1.2]; this should not be surprising, as every effective divisor D on X naturally yields a principal divisor on Spec(Cox(X)), namely, the locus where the associated section s ∈ Γ(X, O_{X}(D)) ⊂ Cox(X) vanishes.

We next exhibit a finitely-generated T_{X}-invariant subalgebra

R ⊂ Cox(X)

such that the induced morphism

j : U →^{ι} Spec(Cox(X)) → Spec(R)

is an open embedding. Choose ample divisors A_{1}, …, A_{r} freely generating Pic(X). (Since being ample is an open condition, we can certainly produce these.) For each ample A_{i}, we obtain an embedding

X ↪ ℙ(w_{i})

into a weighted projective space, where the weights

w_{i} = (w_{i1}, …, w_{ij(i)})

index the degrees of a minimal set of homogeneous generators for the graded ring

x_{i1}, …, x_{ij(i)} ∈ ⨁_{N≥0} Γ(X, O_{X}(NA_{i})).

Take products to obtain

X ↪ ∏_{i=1}^{r} ℙ(w_{i})

and let R denote the multihomogeneous coordinate ring of X, i.e., the quotient of the polynomial ring in the x_{ij} by the multihomogeneous polynomials cutting out X. We can then identify

U = Spec(R) − ⋃_{i=1}^{r}{x_{i1} = ⋯ = x_{ij(i)} = 0}.

Thus we have a diagram

U →^{ι} V := Spec(Cox(X))
j ↓  ↓
j(U) ⊂ W := Spec(R)

with V normal and j an open embedding. Let U′ ⊂ V denote the pre-image of j(U) in V. The induced morphism

β : U′ → j(U) ≃ U

is a birational morphism from a normal variety with a section, induced by ι ∘ j^{−1}. Any such morphism is an isomorphism. Indeed, the composed morphism

U′ →^{β} U →^{ι} U′

agrees with the identity on a dense subset of U′, hence is the identity. Thus β and ι are inverses of each other. □

Theorem 5.8. Let X be a smooth projective surface with −K_{X} nef and big. Then Cox(X) is finitely generated.

Proof. Proposition 4.9 implies that \overline{NE}_{1}(X) is a finite rational polyhedral cone admitting a finite number of (−1)- and (−2)-curves. Thus the nef cone of X takes the form

⟨A_{1}, …, A_{r}⟩

where the A_{i} are nef divisors. Theorem 4.5 guarantees that each A_{i} is semiample. Consider the subring of the Cox ring

Cox′(X) := ⨁_{D∈⟨A_{1},...,A_{r}⟩} Γ(X, O_{X}(D))

which is finitely generated by Proposition 5.5.

We next set up some notation, relying on the fact that −K_{X} is semiample with associated contraction σ : X → S (Corollary 4.6). Let E_{1}, …, E_{r} denote the (−2)- curves on X, i.e., the curves contracted by σ. Let F_{1}, …, F_{s} denote the (−1)-curves on X. Choose generators η_{i} ∈ Γ(O_{X}(E_{i})) and ξ_{i} ∈ Γ(O_{X}(F_{i})), which are unique up to scalars. We regard these as elements of Cox(X).

Lemma 5.9. Let D be an effective divisor on X. Express

(5.3)  D = M + F

where F is the fixed part and M is the moving part. Then the support of F consists of (−1)- and (−2)-curves.

Proof. Suppose that the fixed part of F contains an irreducible component C that is not a (−1)- or (−2)-curve. It follows that C^{2} ≥ 0. Since C is effective, we have

h^{2}(O_{X}(C)) = h^{0}(O_{X}(K_{X} − C)) = 0.

Otherwise, n(K_{X} − C) would be effective for each n ≥ 0, which contradicts our assumption that −K_{X} is big. The Hodge index theorem implies −K_{X} · C > 0, so Riemann-Roch implies h^{0}(O_{X}(C)) > 1, which means that C is not fixed. □

We interpret this via the Cox ring: Each homogeneous element t ∈ Cox(X) can be identified with an effective divisor D = {t = 0}. Expression (5.3) translates into t = m · f, where m ∈ Cox′(X) and

f = η_{1}^{a_{1}} ⋯ η_{r}^{a_{r}}ξ_{1}^{b_{1}} ⋯ ξ_{s}^{b_{s}}, a_{1}, …, a_{r}, b_{1}, …, b_{s} ∈ ℕ.

It follows then that

Cox(X) = Cox′(X)[η_{1}, …, η_{r}, ξ_{1}, …, ξ_{s}]

which completes our proof. □

Remark 5.10. We make a few observations on the significance of Theorem 5.8 and recent generalizations.

• Hu and Keel [HK00] showed that smooth projective varieties with finitely generated Cox rings behave extremely well from the standpoint of birational geometry. Indeed, they designate such varieties Mori Dream Spaces.

• Shokurov [Sho96, §6] demonstrated how a robust version of the log minimal model program would imply that many classes of varieties have finitely generated Cox rings. For example, he established that log Fano threefolds over fields of characteristic zero have this property. These are a natural generalization of the singular Del Pezzo surfaces discussed here.

• As an application of their proof of the existence of minimal models for varieties of log general type (over fields of characteristic zero), Birkar, Cascini, Hacon, and M^{c}Kernan proved that log Fano varieties of arbitrary dimension have finitely generated Cox rings [BCHM06, 1.3.1].

Exercises. Suppose D is an effective divisor on a smooth projective surface X. Consider the graded ring

R(D) := ⨁_{m≥0} Γ(X, O_{X}(mD)).

In the classic paper [Zar62], Zariski analyzed when this ring is finitely generated.

Exercise 5.4.1. Recall the notation of the second half of Exercise 4.4.1. Show that R(D′) is not finitely generated.

Exercise 5.4.2. Assume D admits a Zariski decomposition [Zar62, 7.7] [Laz04, 2.3.19], i.e.,

(5.4)  D = P + N

where P and N are ℚ-divisors with the following properties:

• P is nef;

• N is effective with support

supp(N) = {C_{i}}

generating a negative definite (or trivial) sublattice of the Néron-Severi group;

• P · C_{i} = 0 for each C_{i} ∈ supp(N). Deduce that

• for each n ≥ 0 the map

Γ(X, O_{X}(nD − ⌈nN⌉)) ↪ Γ(O_{X}(nD))

is an isomorphism;

• Γ(X, O_{X}(nP)) ≃ Γ(O_{X}(nD)) for n ≥ 0 such that nN is integral. If −K_{X} is nef and big, deduce also that

• P is semiample;

• supp(N) ⊂ {E_{1}, …, E_{r}, F_{1}, …, F_{s}}, the union of the (−1)- and (−2)-

curves on X.

Hint: The second assertion is a corollary of the first. To prove this, note that any divisor A with

nD − ⌈nN⌉ ≺ A ⪯ nD

intersects some component in supp(N) negatively, and thus has that component in its fixed part.

Exercise 5.4.3. Let X be the Hirzebruch surface 𝔽_{2}, Σ the class of a section at infinity, f the class of a fiber:

\begin{aligned}& Σ & f \\ Σ & 2 & 1 \\ f & 1 & 0 \end{aligned}

This admits a unique (−2)-curve E = Σ − 2f.

• Show that Cox(X) ≃ k[η, f_{0}, f_{∞}, t] where

Γ(O_{X}(E)) = kη, Γ(O_{X}(f)) = kf_{0} + kf_{∞},

and

Γ(O_{X}(Σ)) = kηf_{0}^{2} + kηf_{0}f_{∞} + kηf_{∞}^{2} + kt.

• Show that the Zariski decomposition of the divisor D = Σ − f is

D = P + N, P = \frac{1}{2}Σ, N = \frac{1}{2}E.

Verify that the fixed part of nD for n ≥ 0 is ⌈nN⌉ = ⌈n/2⌉E.

5.5. More Cox rings of Del Pezzo surfaces. For blow-ups β : X → ℙ^{2}, we write L for the pullback of the line class on ℙ^{2} and E_{1}, E_{2}, … for the exceptional curves.

Example 5.11 (Degree Six Del Pezzo Surfaces). Let X be isomorphic to ℙ^{2} blown up at three non-collinear points, which can be taken to be p_{1} = [1, 0, 0], p_{2} = [0, 1, 0], and p_{3} = [0, 0, 1]. This is a toric variety under the action of the diagonal torus. We have seen in §2.2 that \overline{NE}_{1}(X) is generated by the (−1)-curves:

{E_{1}, E_{2}, E_{3}, E_{12}, E_{13}, E_{23}}

where E_{ij} is the proper transform of the line joining p_{i} and p_{j} with class L−E_{i}−E_{j}. Here we have (cf. [BP04, 3.1]):

Cox(X) = k[η_{1}, η_{2}, η_{3}, η_{12}, η_{13}, η_{23}].

Example 5.12 (Degree Five Del Pezzo Surfaces). This example is due to Skorobogatov [Sko93] (see also [BP04, 4.1]). Suppose that X is isomorphic to ℙ^{2} blown up at four points in linear general position, which can be taken to be p_{1} = [1, 0, 0], p_{2} = [0, 1, 0], p_{3} = [0, 0, 1], and p_{4} = [1, 1, 1]. Let E_{i}, i = 1, …, 4 denote the exceptional curves and E_{ij} the proper transforms of the lines joining p_{i}, with class E_{ij} = L − E_{i} − E_{j}. Skorobogatov shows there exist normalizations of the generators η_{i5} ∈ Γ(O_{X}(E_{i})) and η_{ij} ∈ Γ(O_{X}(E_{ij})) such that

Cox(X) = k[η_{12}, …, η_{45}]/ ⟨P_{1}, P_{2}, P_{3}, P_{4}, P_{5}⟩

where each P_{i} is a Plücker relation

P_{i} = η_{jk}η_{lm} − η_{jl}η_{km} + η_{jm}η_{kl}, {i, j, k, l, m} = {1, 2, 3, 4, 5}, j < k < l < m.

More geometrically, Cox(X) is the projective coordinate ring of the Grassmannian 𝔾(1, 4) ⊂ ℙ^{9}.

Example 5.13 (E_{6} cubic surface). See [HT04, §3] for more details. Let S ⊂ ℙ^{3} denote the (unique) cubic surface with a singularity of type E_{6}

S = {(w, x, y, z) : xy^{2} + yw^{2} + z^{3} = 0} ⊂ ℙ^{3}

and σ : X → S its minimal resolution of singularities. Let E_{1}, …, E_{6} denote the exceptional curves of σ and ℓ ⊂ X the proper transform of the unique line {y = z = 0} ⊂ S. The effective cone here is simplicially generated by (−1)- and (−2)-curves

\overline{NE}_{1}(X) = ⟨ℓ, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}⟩

but the corresponding elements ξ_{ℓ}, ξ_{1}, …, ξ_{6} ∈ Cox(X) do not suffice to generate it. In this case, for a suitable ordering of the E_{i} we have

Cox(X) ≃ k[ξ_{1}, ..., ξ_{6}, ξ_{ℓ}, τ_{1}, τ_{2}, τ_{ℓ}]/⟨τ_{ℓ}ξ_{ℓ}^{3}ξ_{4}^{2}ξ_{5} + τ_{2}^{2}ξ_{2} + τ_{1}^{3}ξ_{1}^{2}ξ_{3}⟩.

We mention some other significant results:

• Batyrev and Popov [BP04] showed that the Cox ring of a Del Pezzo surface X of degree d = 2, 3, 4, 5, 6 is generated by sections associated with (−1)-curves on X. They show the relations (up to radical) are given by quadratic expressions analogous to the Plücker-type relations above. Furthermore, they conjectured that these quadratic relations actually generate the ideal of all relations.

• The Batyrev-Popov conjecture was proven for Del Pezzo surfaces of degree d ≥ 4 and cubic surfaces without Eckardt points by Stillman, Testa, and Velasco [STV07]. Derenthal [Der06a] has also made significant contributions to our understanding of the relations in the Cox ring.

• Laface and Velasco [LV07] established the Batyrev-Popov conjecture when d ≥ 2. Sturmfels and Xu [SX08] and Testa, Várilly-Alvarado, and Velasco [TVAV08] address Del Pezzo surfaces of degree one.

• For d = 2, 3, 4, 5 the affine variety Spec(Cox(X)) can be related to homogeneous spaces G/P, where G is a simply-connected algebraic group associated to the root system arising from K_{X}^{⊥} ⊂ N^{1}(S, ℤ) (cf. §3.3.) Here P is the maximal parabolic subgroup associated to a representation of G naturally connected with the (−1)-curves on X. (This generalized the relation discussed between Grassmannians and Cox rings of degree-five Del Pezzos.) See [SS07] and [Der07] for details, as well as [Pop01] for the case of degree four.

• There are numerous examples of singular Del Pezzo surfaces (like the E_{6} cubic surface) whose Cox rings admit a single relation. These are classified in [Der06b].

Exercises.

Exercise 5.5.1. Let X be the blow-up of ℙ^{2} at three collinear points. Compute generators and relations for Cox(X). Hint: You can find the answer in [Has04].

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Department of Mathematics, Rice University, Houston, Texas 77005, USA

    E-mail address: hassett@math.rice.edu