QUARTIC DEL PEZZO SURFACES OVER FUNCTION FIELDS OF CURVES

BRENDAN HASSETT AND YURI TSCHINKEL

1. Introduction

The geometry of spaces of rational curves of low degree on Fano threefolds is a very active area of algebraic geometry. One of the main goals is to understand the Abel-Jacobi morphism from the base of a space of such curves to the intermediate Jacobian of the threefold. For instance, does a given space of curves dominate the intermediate Jacobian? If so, what are the fibers of this morphism? Does it give the maximal rationally connected (MRC) quotient? Representative results in this direction are available for:

• cubic threefolds [HRS05, JHS, IM00, MT01];

• Fano threefolds of genus six and degree 10 in ℙ^{7} [DIM12];

• Fano threefolds of genus seven and degree 12 in ℙ^{8} [IM07];

• moduli of vector bundles [Cas04]—this case makes clear that one cannot always expect the morphism to the intermediate Jacobian to give the MRC fibration.

At the same time, del Pezzo fibrations π : X → ℙ^{1} are equally interesting geometrically. Moreover, the special case where the rational curves happen to be sections of π is particularly important for arithmetic applications. It is a major open problem to determine whether or not sections exist over a non-closed ground field. Of course, the Tsen-Lang theorem gives sections when the ground field is algebraically closed. Even when there are rigid sections, these are typically defined over extensions of large degree.

However, suppose that the space of sections of fixed height is rationally connected over the intermediate Jacobian IJ(X). If the fibration is defined over a finite field and the space of sections descends to this field then π has sections over that field. Indeed, this follows by combining a theorem of Lang [Lan55] (principal homogeneous spaces for abelian varities over finite fields are trivial) and a theorem of Esnault [Esn03] (rationally connected varieties over finite fields have rational points). This point of view was developed for quadric surface bundles in [HT12b], with applications to effective versions of weak approximation.

Date: July 14, 2013.

This paper addresses the case of quartic del Pezzo fibrations π : X → ℙ^{1}. Throughout, we assume X is smooth and the degenerate fibers of π have at worst one ordinary singularity. We classify numerical invariants—the fundamental invariant is the height of the fibration, denoted h(X). We provide explicit geometric realizations for fibrations of small height. We exhibit families of sections parametrizing the intermediate Jacobian and admitting rationally-connected fibrations over the intermediate Jacobian. The underlying constructions often involve Brill-Noether theory for smooth or nodal curves. Our main result is the comprehensive analysis of spaces of small-height sections for height 12 fibrations in Section 10; here X dominates a nodal Fano threefold of genus seven and degree 12.

Acknowledgments: The first author was supported by National Science Foundation Grants 0901645, 0968349, and 1148609; the second author was supported by National Science Foundation Grants 0739380, 0968318, and 1160859. We are grateful to Asher Auel, Jean-Louis Colliot-Thélène, and R. Parimala for useful conversations as well as to Andrew Kresch and Nikita Kozin for comments on the manuscript; we appreciate Jason Starr and Yi Zhu explaining the results of [Zhu11] on Abel-Jacobi morphisms for fibrations of rationally connected varieties over curves.

2. Basic properties of quartic del Pezzo surfaces

Let X be a quartic del Pezzo surface over an algebraically closed field. X is isomorphic to ℙ^{2} blown up at five distinct points such that no three are collinear. Thus we may identify

Pic(X) = ℤL + ℤE_{1} + ℤE_{2} + ℤE_{3} + ℤE_{4} + ℤE_{5},

where L is the pull-back of the hyperplane class on ℙ^{2} and the E_{i} are the exceptional divisors. The primitive divisors

Λ = K_{X}^{⊥} ⊂ Pic(X)

are a lattice under the intersection form; using the basis

{E_{1} − E_{2}, E_{2} − E_{3}, E_{3} − E_{4}, E_{4} − E_{5}, L − E_{1} − E_{2} − E_{3}}

we have

Λ ≃ (\begin{matrix}−2 & 1 & 0 & 0 & 0 \\ 1 & −2 & 1 & 0 & 0 \\ 0 & 1 & −2 & 1 & 1 \\ 0 & 0 & 1 & −2 & 0 \\ 0 & 0 & 1 & 0 & −2 \end{matrix}) .

Writing Λ^{∗} = Hom(Λ, ℤ), the intersection form gives a surjection

Pic(X) ↠ Λ^{∗},

with kernel generated by K_{X}. Note that Λ^{∗}/Λ is a cyclic group of order four.

The action of the monodromy group on Pic(X) factors through the Weyl group W (D_{5}) coming from the D_{5} root system contained in the lattice (−1)Λ. Abstractly, W (D_{5}) may be realized as an extension of the symmetric group

(2.1)  1 → (ℤ/2ℤ)^{4} → W (D_{5}) → 𝔖_{5} → 1;

concretely, this is the subgroup of the group of signed 5×5 permutation matrices where the determinant of the matrix equals the sign of the permutation.

The anticanonical model of X is a complete intersection of two quadrics

X = {Q_{0} = Q_{1} = 0} ⊂ ℙ^{4}.

Let Gr(2, Γ(O_{ℙ^{4}}(2))) ≃ Gr(2, 15) denote the Grassmannian parametrizing pencils of quadrics in ℙ^{4}. The group SL_{5} acts naturally on ℙ^{4} and this Grassmannian, linearized via the Plücker embedding. Consider the loci of stable and semistable points

U_{s} ⊂ U_{ss} ⊂ Gr(2, Γ(O_{ℙ^{4}}(2)))

and the resulting GIT quotient

\overline{M} := Gr(2, Γ(O_{ℙ^{4}}(2)))//SL_{5}.

Principal results of [MM93] include:

• U_{s} coincides with the locus of smooth quartic del Pezzo surfaces;

• U_{ss} parametrizes quartic del Pezzo surfaces with at worst ordinary double points;

• smooth quartic del Pezzo surfaces are diagonalizable, i.e., there exist coordinates x_{0}, …, x_{4} on ℙ^{4} such that

X = {x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = c_{0}x_{0}^{2} + c_{1}x_{1}^{2} + c_{2}x_{2}^{2} + c_{3}x_{3}^{2} + c_{4}x_{4}^{2} = 0}; • closed orbits in U_{ss} correspond to diagonalizable del Pezzo sur-

faces, which have 0, 2, or 4 ordinary double points;

• the determinantal scheme of a quartic del Pezzo surface

X ↦ Z = {det(s_{0}Q_{0} + s_{1}Q_{0}) = 0} ⊂ ℙ_{[s_{0},s_{1}]}^{1} = ℙ(Γ(I_{X}(2)))

induces a rational map

(2.2)  Ξ : Gr(2, Γ(O_{ℙ^{4}}(2))) −−→ ℙ(Γ(O_{ℙ^{1}}(5)))//SL_{2},

regular on U_{ss};

• Ξ takes smooth del Pezzo surfaces to binary quintics with distinct roots; it takes surfaces with at most two ordinary double points to binary quintics with at most one double root; in any event, the resulting binary quintic has at worst double roots;

• we obtain an isomorphism of projective schemes

\overline{M} ≃ ℙ(Γ(O_{ℙ^{1}}(5)))//SL_{2}.

The diagonal representation of a smooth quartic del Pezzo surface has useful implications:

Consider the group H = (ℤ/2ℤ)^{4}, realized as 5 × 5 diagonal matrices with ±1 on the diagonal modulo the subgroup ±I. Then H acts on X and this action is transitive on the 16 lines of X.

Moreover, the generic quartic del Pezzo surface has automorphism group H [Koi88, p. 131].

The del Pezzo surfaces with > 2 ordinary double points are mapped to the distinguished point s corresponding to the binary quintic form with two double roots. The invariants of binary quintics were computed in the 19th century; see [Bak08] for a description convenient for our purposes. It follows that

\overline{M} ≃ ℙ(1, 2, 3)

and the minimal degree invariant is non-zero at s ∈ ℙ(1, 2, 3). We may interpret these GIT quotients as \overline{M}_{0,5}/𝔖_{5}, where the symmetric group acts by permuting the points.

Proposition 1. Let M denote the moduli stack of quartic del Pezzo surfaces with at worst one ordinary double point and write M = \overline{M} \ s. The morphism Ξ (defined in (2.2)) induces a morphism

ξ : M → M

that is bijective on geometric points.

Proof. The closure in U_{ss} of the orbit of any quartic del Pezzo surface with a unique ordinary double point contains quartic del Pezzo surfaces with at least two ordinary double points. Indeed, quartic del Pezzo surfaces with one ordinary double point are strictly semistable but have orbits that are not closed. In particular, the moduli stack of quartic del Pezzo surfaces with ordinary singularities is non-separated. Concretely, fix a quartic del Pezzo surface X = {Q_{0} = Q_{1} = 0} with one ordinary singularity. Supposing the singularity is at [0, 0, 0, 0, 1] with tangent space x_{0} = 0, and also that Q_{0} cuts out this tangent space and Q_{1} is nodal at [0, 0, 0, 0, 1], we obtain

Q_{0} = x_{0}x_{4} + Q_{2}(x_{0}, x_{1}, x_{2}, x_{3}), Q_{1} = Q_{1}(x_{0}, x_{1}, x_{2}, x_{3}).

Writing Q_{2} = x_{0}ℓ_{2}(x_{0}, x_{1}, x_{2}, x_{3}) + R_{2}(x_{1}, x_{2}, x_{3}) and substituting x_{4} = x′_{4} − ℓ_{2}, we obtain

X = {x′_{4}x_{0} + R_{2}(x_{1}, x_{2}, x_{3}) = Q_{1} = 0}.

Writing Q_{1} = x_{0}ℓ_{1}(x_{0}, x_{1}, x_{2}, x_{3}) + R_{1}(x_{1}, x_{2}, x_{3}) yields

X = {x′_{4}x_{0} + R_{2}(x_{1}, x_{2}, x_{3}) = x_{0}ℓ_{1} + R_{1}(x_{1}, x_{2}, x_{3}) = 0}.

Consider the action of the one-parameter subgroup ρ(t) given by

x_{0} ↦ tx_{0}, x_{i} ↦ x_{i}, i = 1, 2, 3, x′_{4} ↦ t^{−1}x′_{4}.

The limit

X_{0} := lim_{t→0}ρ(t) · X = {x′_{4}x_{0} + R_{2}(x_{1}, x_{2}, x_{3}) = R_{1}(x_{1}, x_{2}, x_{3}) = 0}

has double points [0, 0, 0, 0, 1] and [1, 0, 0, 0, 0]. Moreover, when {R_{1} = R_{2} = 0} ⊂ ℙ^{2} consists of four distinct points we have

Aut^{∘}(X_{0}) ≃ 𝔾_{m},

i.e., the identity component of the automorphism group corresponds to ρ(t).

It is a fundamental result of Geometric Invariant Theory that orbit specializations are governed by specializations under one-parameter subgroups arising as automorphisms of objects with closed orbit (see [HH13, Prop. 4.2]). Note that there is a unique non-closed orbit in the basin of attraction of X_{0} under ρ(t) (resp. under ρ^{−1}(t)). Furthermore, there is an involution ι of X_{0}

[x_{0}, x_{1}, x_{2}, x_{3}, x′_{4}] ↦ [x′_{4}, x_{1}, x_{2}, x_{3}, x_{0}]

whose action on U_{ss} exchanges these non-closed 𝔾_{m}-orbits. Thus there is a single isomorphism class of del Pezzo surfaces in M with X_{0} as a one-parameter limit. □

Remark 2 (suggested by Andrew Kresch). The stack M is nevertheless non-separated: If M were separated it would have finite diagonal, which means that the order of the stabilizer group could only increase under specialization. A generic smooth del Pezzo surface has automorphism group (ℤ/2ℤ)^{4}. However, a generic del Pezzo surface with a single node has automorphism group of order eight [Hos96, p. 375].

A consequence of Proposition 1 is

Corollary 3. Pic(M) ≃ ℤ.

3. Quartic del Pezzo fiber spaces over ℙ^{1}

We work over an algebraically closed field of characteristic different from two.

Proposition 4. Let X be a smooth projective threefold and π : X → ℙ^{1} a morphism such that the generic fiber is a smooth rational surface. Then

• X is separably rationally connected;

• the Hodge numbers h^{ij}(X) = 0 for (i, j) ≠ (0, 0), (1, 1), (2, 2), (3, 3), (1, 2), (2, 1).

Proof. π admits a section s : ℙ^{1} → X. This follows from general results [dJS03] or by the classification of rational surfaces and the Tsen-Lang theorem (cf. [Has09]). Since the smooth fibers of π are rational surfaces, there exist very free curves in the fibers incident to s(ℙ^{1}). The techniques of [GHS03] thus yield a very free section of π, proving separably rational connectedness.

It follows [Kol96, IV.3.8] that

Γ(Ω_{X}^{1} ) = Γ(Ω_{X}^{2} ) = Γ(Ω_{X}^{3} ) = 0

and by Serre duality

H^{3}(Ω_{X}^{2} ) = H^{3}(Ω_{X}^{1} ) = H^{3}(O_{X}) = 0.

The Leray spectral sequence for π and Serre duality also give

H^{1}(O_{X}) = H^{2}(O_{X}) = 0 H^{2}(Ω_{X}^{3} ) = H^{1}(Ω_{X}^{3} ) = 0.
□

Assume that π : X → ℙ^{1} is a fibration of quartic del Pezzo surfaces, with X smooth. The fibration has square-free discriminant if the fibers are complete intersections of two quadrics with at worst one ordinary double point; we assume this from now on. Under these assumptions, the relative dualizing sheaf ω_{π} is invertible, the direct image of its dual

π_{∗}ω_{π}^{−1}

is locally free, and the higher direct images vanish. Cohomology-and-base-change and the fact that ω_{π}^{−1} is very ample on fibers of π yield an embedding X ⊂ ℙ((π_{∗}ω_{π}^{−1})^{∨}). We have that

π_{∗}(I_{X} ⊗ O_{ℙ((π_{∗}ω_{π}^{−1})^{∨})}(2)) ⊂ Sym^{2}(π_{∗}ω_{π}^{−1})

is a subbundle of rank two with quotient π_{∗}ω_{π}^{−2}.

Proposition 5. Let π : X → ℙ^{1} be a fibration of quartic del Pezzo surfaces with square-free discriminant, defined over a field of characteristic p > 2. Then X and π are liftable to characteristic zero.

Remark 6. Assuming Proposition 5, the Hodge-de Rham spectral sequence for X degenerates provided p > 3 (or = 0) [DI87]. Thus h^{11}(X) equals the Picard rank of X, and h^{11} and h^{12} are unchanged on reduction to X from characteristic zero.

Proof. Every vector bundle over ℙ^{1} decomposes as a direct sum of line bundles

V ≃ ⊕_{i=1}^{r}O_{ℙ^{1}}(a_{i}).

Clearly any such direct sum lifts to characteristic zero, i.e., lift each summand individually. Cohomology commutes with reduction modulo p for such a lift; given lifts of two such bundles, taking homomorphisms between them commutes with reduction as well.

In particular, we may lift the decomposition of V = π_{∗}ω_{π}^{−1} to characteristic zero; the same holds true for Sym^{2}V and its quotient V′ = π_{∗}ω_{π}^{−2}. There is a combinatorial criterion [Sha78] for which vector bundles arise as subbundles of a given bundle, independent of the characteristic. In particular, the quotient homomorphism

Sym^{2}V ↠ V′

lifts to characteristic zero, as does its kernel. The resulting rank-two subbundle gives the defining equations of the lift of X to characteristic zero. Having square-free discriminant is an open condition, so the resulting lift shares this property. □

We define

h(X) := deg(c_{1}(ω_{π})^{3})

to be the height of the del Pezzo surface fibration. Corollary 3 implies this is the unique numerical invariant for families with square-free discriminant:

Proposition 7. Let π : X → ℙ^{1} be a quartic del Pezzo fibration with square-free discriminant. Then we have:

• h(X) = −2 deg(π_{∗}ω_{π}^{−1});

• Δ := number of singular fibers of π = 2h(X);

• the topological Euler characteristic

χ(X) = (2 − Δ) · χ(smooth fiber) + Δ · χ(singular fiber)
= 16 − 2h(X);

• χ(Ω_{X}^{1} ) = 1 − χ(X)/2 = h(X) − 7;

• expected number of parameters = −χ(T_{X}) = \frac{3}{2}h(X) − 1.

Proof. The quantities h(X), deg(π_{∗}ω_{π}^{−1}), and Δ all depend on the degree of the classifying map μ : ℙ^{1} → M, thus are in constant proportion. To evaluate the proportionality constant, it suffices to consider a singular non-constant example, e.g., the intersection of a quadric in ℙ^{4} with a pencil of quadric hypersurfaces. (See Case 1 of Section 4 for details.)

Once we know the number of singular fibers, it is straightforward to compute the (étale) topological Euler characteristic. The Euler characteristics of Ω_{X}^{1} and Ω_{X}^{2} are related to the topological Euler characteristic via the Hodge-de Rham spectral sequence

χ(X) = χ(O_{X}) − χ(Ω_{X}^{1} ) + χ(Ω_{X}^{2} ) − χ(Ω_{X}^{3} );

Serre duality and Proposition 4 give

χ(X) = 2 − 2χ(Ω_{X}^{1} ).

Finally, the expected number of parameters −χ(T_{X}), which may be computed by Hirzebruch-Riemann-Roch and Gauss-Bonnet:

\begin{aligned}χ(T_{X}) & = & \frac{1}{2}c_{1}(T_{X})^{3} − \frac{19}{24}c_{1}(T_{X})c_{2}(T_{X}) + \frac{1}{2}c_{3}(T_{X}) \\ & = & −\frac{1}{2}c_{1}(ω_{X})^{3} − 19χ(O_{X}) + \frac{1}{2}χ(X). \end{aligned}

We have

c_{1}(ω_{X})^{3} = (c_{1}(π^{∗}ω_{ℙ^{1}}) + c_{1}(ω_{π}))^{3} = h(X) − 24

whence

χ(T_{X}) = −\frac{1}{2}h(X) + 12 − 19 + 8 − h(X) = −\frac{3}{2}h(X) + 1,

as claimed. □

Combining with Proposition 4 and Remark 6, we obtain

Corollary 8. Retain the notation and assumptions of Proposition 7; assume the characteristic p > 3 (or = 0) and set

h^{11} = dim H^{1}(Ω_{X}^{1} ) = rank(Pic(X)).

Then the Hodge diamond of X takes the form

1

0  0
0  h^{11}  0
0  h(X) + h^{11} − 7  h(X) + h^{11} − 7  0

This can be interpreted in geometric terms: Let π : X → ℙ^{1} be a fibration of quartic del Pezzo surfaces with square-free discriminant. Let D → ℙ^{1} be the degree-five covering parametrizing nodal quadric hypersurfaces containing the fibers of π, and D̃ → D the double covering parametrizing rulings of these quadric hypersurfaces. The assumption on singular fibers implies that D → ℙ^{1} is simply branched and D̃ → D is étale, and we may consider the resulting Prym variety Prym(D̃ → D).

Proposition 9. Retaining the notation above, let A^{2}(X) denote codimension two cycles algebraically equivalent to zero. Then A^{2}(X) ≃ Prym(D̃ → D).

Proof. Fix a section s : ℙ^{1} → X of π, not passing through any lines of the singular fibers. Projecting from s realizes the generic fiber of π as a conic bundle over ℙ^{1}. Indeed, blowing up a quartic del Pezzo surface at a point not on a line yields a cubic surface with a line; projecting the cubic surface from that line gives a conic bundle over ℙ^{1}. There are five degenerate fibers, corresponding to the generic fiber of D → ℙ^{1}, and ten irreducible components in these fibers, corresponding to the the generic fiber of D̃ → ℙ^{1}. Thus the blow-up of X along s(ℙ^{1}) is a conic bundle over a Hirzebruch surface surface, with discriminant curve D and étale cover D̃ → D. A straightforward variation on the argument of [Bea77, §3] implies that A^{2}(Bl_{s(ℙ^{1})}(X)) ≃ Prym(D̃ → D); the same holds for A^{2}(X). (For related results, see [Kan89].) □

Remark 10. Over ℂ, this Prym variety is the intermediate Jacobian IJ(X); its construction is compatible with reduction modulo p, assuming the reduction still has square-free discriminant. Even over fields of characteristic p > 0, Prym(D̃ → D) has the requisite universal property for one-cycles [Bea77, §3.2]: Given a space of curves

\begin{matrix}C & → & X \\ ↓ \\ T \end{matrix}

over a smooth connected base T, there is an Abel-Jacobi morphism T → Prym(D̃ → D).

The Lefschetz hyperplane theorem gives:

Proposition 11. Let X → ℙ^{1} be a fibration of quartic del Pezzo surfaces. Suppose X ⊂ ℙ((π_{∗}ω_{π}^{−1})^{∨}) is a smooth complete intersection of ample divisors in ℙ((π_{∗}ω_{π}^{−1})^{∨}). Then Pic(X) = ℤ^{2}.

Definition 12. Let π : X → ℙ^{1} be a quartic del Pezzo fibration with square-free discriminant. The anticanonical height of a section σ : ℙ^{1} → X is defined

h_{ω_{π}^{−1}}(σ) = deg(σ^{∗}ω_{π}^{−1}).

Note that this equals the degree of the normal bundle N_{σ}, thus the Hilbert scheme of sections has dimension ≥ h_{ω_{π}^{−1}}(σ) + 2 at σ.

4. Explicit realization of generic families

In this section, we give concrete geometric realizations of families of each height. We start by writing

\begin{matrix}π_{∗}ω_{π}^{−1} = O_{ℙ^{1}}(−a_{1}) ⊕ O_{ℙ^{1}}(−a_{2}) ⊕ O_{ℙ^{1}}(−a_{3}) ⊕ O_{ℙ^{1}}(−a_{4}) ⊕ O_{ℙ^{1}}(−a_{5}), \\ a_{1} ≤ a_{2} ≤ a_{3} ≤ a_{4} ≤ a_{5}. \end{matrix}

We refer the reader to [Shr06, 2.2] for a discussion of which a_{j} may occur. Here we assume this sheaf is generic, in the sense that it is as ‘balanced’ as possible, i.e., a_{5} − a_{1} ≤ 1. Thus π_{∗}ω_{π}^{−1} is one of the following:

(1) O_{ℙ^{1}}^{5} ⊗ O_{ℙ^{1}}(−m)

(2) (O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−1)^{4}) ⊗ O_{ℙ^{1}}(−m)

(3) (O_{ℙ^{1}}^{2} ⊕ O_{ℙ^{1}}(−1)^{3}) ⊗ O_{ℙ^{1}}(−m)

(4) (O_{ℙ^{1}}^{3} ⊕ O_{ℙ^{1}}(−1)^{2}) ⊗ O_{ℙ^{1}}(−m)

(5) (O_{ℙ^{1}}^{4} ⊕ O_{ℙ^{1}}(−1)) ⊗ O_{ℙ^{1}}(−m) where m ∈ ℤ. In this situation, Proposition 7 may be proven by direct computations with resolutions.

Case 1. We compute invariants for generic quartic del Pezzo surfaces

π : X → ℙ^{1}

arising from morphisms into the Hilbert scheme

μ : ℙ^{1} → Gr(2, Γ(O_{ℙ^{4}}(2)))

of degree d ≥ 0, where μ is an embedding transverse to the discriminant. In this situation, we may embed

X ⊂ P := ℙ^{1} × ℙ^{4},

where

(4.1)  π_{∗}(I_{X} (0, 2)) ⊂ Γ(O_{ℙ^{4}}(2)) ⊗ O_{ℙ^{1}}

is a rank-two vector subbundle of degree −d.

Even part. Assuming this bundle is generic and d = 2n, we have

π_{∗}(I_{X} (0, 2)) ≃ O_{ℙ^{1}}(−n)^{2}

and X is a complete intersection of two forms of bidegree (n, 2). As such, X depends on

2(15(n + 1) − 2) − (3 + 24) = 30n − 1

parameters. In this situation

ω_{π}^{−1} = O_{X} (−2n, 1), π_{∗}ω_{π}^{−1} = O_{ℙ^{1}}(−2n) ⊗ Γ(O_{ℙ^{4}}(1))

whence

deg(π_{∗}ω_{π}^{−1}) = −10n, h(X) = 20n.

We compute the middle cohomology using the exact sequence

0 → I_{X} /I_{X}^{2} → Ω_{P}^{1} |X → Ω_{X}^{1} → 0,

where the conormal bundle

I_{X} /I_{X}^{2} = O_{X} (−n, −2)^{2}.

Thus we find

χ(Ω_{X}^{1} ) = χ(Ω_{P}^{1} |X) − 2χ(O_{X} (−n, −2)).

Using the Koszul resolution

0 → O_{P} (−2n, −4) → O_{P} (−n, −2)^{2} → I_{X} → 0

we compute

\begin{aligned}χ(O_{X} (a, b)) & = & χ(O_{P} (a, b)) − 2χ(O_{P} (a − n, b − 2)) + χ(O_{P} (a − 2n, b − 4)) \\ χ(Ω_{P}^{1} |X (a, b)) & = & χ(Ω_{P}^{1} (a, b)) − 2χ(Ω_{P}^{1} (a − n, b − 2)) + χ(Ω_{P}^{1} (a − 2n, b − 4)). \end{aligned}

The formulas

χ(O_{P} (a, b)) = (a + 1)\binom{b + 4}{4},

χ(Ω_{P}^{1} (a, b)) = (a − 1)\binom{b + 4}{4} + (5\binom{b + 3}{4} − \binom{b + 4}{4}) (a + 1),

thus give

χ(Ω_{X}^{1} ) = −7 + 20n, χ(X) = 16 − 40n.

When n > 0 we obtain

h^{2}(Ω_{X}^{1} ) = −5 + 20n

as well.

Remark 13. The special case n = 0 is a constant fibration of quartic del Pezzo surfaces, which clearly has sections.

Odd part. Now assume d = 2n + 1 so (4.1) takes the form

π_{∗}I_{X} (0, 2) ≃ O_{ℙ^{1}}(−n) ⊕ O_{ℙ^{1}}(−n − 1)

and X is a complete intersection of forms of bidegrees (n, 2) and (n + 1, 2). In this situation

ω_{π}^{−1} = O_{X} (−2n − 1, 1), π_{∗}ω_{π}^{−1} = O_{ℙ^{1}}(−2n − 1) ⊗ Γ(O_{ℙ^{4}}(1))

whence

deg(π_{∗}ω_{π}^{−1}) = −10n − 5, h(X) = 20n + 10.

Remark 14. The base case n = 0 is studied in Section 9, where we show that it always admits a section σ : ℙ^{1} → X, defined over the coefficient field.

Case 2. We next consider

X ⊂ P := ℙ^{1} × ℙ^{5}

contained in a complete intersection of a form L of bidegree (1, 1).

Even part. Consider when X is a complete intersection of a form L of bidegree (1, 1) and two forms of bidegree (n, 2). Let V be defined by the exact sequence

0 → O_{ℙ^{1}}(−1) →^{·L} Γ(O_{ℙ^{5}}(1)) ⊗ O_{ℙ^{1}} → V → 0.

For generic L, we have

V ≃ O_{ℙ^{1}}^{4} ⊕ O_{ℙ^{1}}(1).

We have

ω_{π}^{−1} = O_{X} (−2n − 1, 1)

and the projection formula yields

π_{∗}ω_{π}^{−1} = O_{ℙ^{1}}(−2n − 1) ⊗ V,

whence

deg(π_{∗}ω_{π}^{−1}) = 4(−2n − 1) − 2n = −10n − 4, h(X) = 20n + 8.

Remark 15. We analyze rational points in the case n = 0. Consider the projection Y := π_{2}(X) ⊂ ℙ^{5} onto the second factor; it is a complete intersection of two quadrics. The fibration corresponds to a pencil of hyperplane sections of Y. Let Z ⊂ Y be its base locus, so that X = Bl_{Z}(Y). Since X is smooth, Z must be a smooth curve of genus one.

The constant subfibration

Z × ℙ^{1} ⊂ X
↓  ↙
ℙ^{1}

has a section over the coefficient field k, provided Z(k) ≠ ∅.

Odd part. Now consider

X ⊂ P := ℙ^{1} × ℙ^{5}

as a complete intersection of a form L of bidegree (1, 1), a form of bidegree (n, 2), and a form of bidegree (n + 1, 2). We have

ω_{π}^{−1} = O_{X} (−2n − 2, 1)

so that

π_{∗}ω_{π}^{−1} = O_{ℙ^{1}}(−2n − 2) ⊗ V,

where V is as above. Thus we conclude

deg(π_{∗}ω_{π}^{−1}) = 4(−2n − 2) + (−2n − 1) = −10n − 9, h(X) = 20n + 18.

Case 3. Consider

X ⊂ ℙ^{1} × ℙ^{6}

as a complete intersection of two forms L_{1} and L_{2} of bidegree (1, 1) and two additional forms quadratic in the second factor. Let V be defined by the exact sequence

0 → O_{ℙ^{1}}(−1)^{2} →^{·(L_{1},L_{2})} Γ(O_{ℙ^{6}}(1)) ⊗ O_{ℙ^{1}} → V → 0;

for generic L_{1} and L_{2}, we have

V ≃ O_{ℙ^{1}}(1)^{2} ⊕ O_{ℙ^{1}}^{3}.

As before, we have

π_{∗}O_{X} (1) ≃ V.

Even part. Suppose that the additional quadratic forms both have degree (n, 2). Then ω_{π} ≃ O_{X} (2n + 2, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 1)^{2} ⊕ O_{ℙ^{1}}(−2n − 2)^{3}

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 8, h(X) = 20n + 16.

Odd part. Suppose that the additional forms have degrees (n, 2) and (n + 1, 2). Then ω_{π} ≃ O_{X} (2n + 3, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 2)^{2} ⊕ O_{ℙ^{1}}(−2n − 3)^{3}

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 13, h(X) = 20n + 26.

Example 16 (n = −1 case). We make sense of the n = −1 case using a slightly different construction. The projective bundle

ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{2} ⊕ O_{ℙ^{1}}^{3}) = {L_{1} = L_{2} = 0} ⊂ ℙ^{1} × ℙ^{6}

admits effective divisors that are not restrictions of effective divisors from the product. We exploit these in our construction. Specifically,

Γ(O_{ℙ(V^{∨})}(a) ⊗ π^{∗}O_{ℙ^{1}}(b)) = Γ(Sym^{a}(V) ⊗ O_{ℙ^{1}}(b)),

which is nonzero if and only if a + b ≥ 0. The case a = 2, b = −2 corresponds to reducible quadrics in ℙ(V^{∨}) singular along ℙ(O_{ℙ^{1}}^{3}), so we focus on the case a = 2, b = −1. Here we have

Sym^{2}(V) ⊗ O_{ℙ^{1}}(−1) = O_{ℙ^{1}}(1)^{3} ⊕ O_{ℙ^{1}}^{6} ⊕ O_{ℙ^{1}}(−1)^{6},

i.e., the vector space of global sections has dimension 12; these all vanish along ℙ(O_{ℙ^{1}}^{3}) ⊂ ℙ(V^{∨}). Since non-singular quadric threefolds contain no two-planes, divisors D in O_{ℙ(V^{∨})(2)} ⊗ π^{∗}O_{ℙ^{1}}(−1) are fibered over ℙ^{1} in singular quadric hypersurfaces.

Let X = D ∩ Q where Q has bidegree (0, 2), i.e., Q is the pullback of a quadric from ℙ^{6}. The image of ℙ(V^{∨}) in ℙ^{6} is a quadric hypersurface.

Since X is contained in a singular quadric hypersurface, its monodromy is non-maximal; indeed, W (D_{5}) acts on the five singular quadric hypersurfaces containing X via the canonical homomorphism W (D_{5}) → 𝔖_{5}. This is consistent with Proposition 11.

Remark 17. Let C denote the intersection of the distinguished twoplane in ℙ^{4} with Q; we have a constant subfibration

C × ℙ^{1} ⊂ X .

It follows that X → ℙ^{1} admits sections over any ground field.

Case 4. Consider

X ⊂ ℙ^{1} × ℙ^{7}

as a complete intersection of three forms L_{1}, L_{2}, and L_{3} of bidegree (1, 1) and two additional forms quadratic in the second factor. Let V be defined by the exact sequence

0 → O_{ℙ^{1}}(−1)^{3} →^{·(L_{1},L_{2},L_{3})} Γ(O_{ℙ^{7}}(1)) ⊗ O_{ℙ^{1}} → V → 0;

for generic L_{i}, we have

V ≃ O_{ℙ^{1}}(1)^{3} ⊕ O_{ℙ^{1}}^{2}.

Even part. Suppose that the additional quadratic forms both have degree (n, 2). Then ω_{π} ≃ O_{X} (2n + 3, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 2)^{3} ⊕ O_{ℙ^{1}}(−2n − 3)^{2}

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 12, h(X) = 20n + 24.

Example 18 (n = −1 case). Consider the intersection of the forms L_{1}, L_{2}, and L_{3}:

ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{3} ⊕ O_{ℙ^{1}}^{2}) ⊂ ℙ(O_{ℙ^{1}}^{8}).

In this case, we have

Sym^{2}(V) ⊗ O_{ℙ^{1}}(−1) = O_{ℙ^{1}}(1)^{6} ⊕ O_{ℙ^{1}}^{6} ⊕ O_{ℙ^{1}}(−1)^{3},

i.e., the space of sections has dimension 18. Furthermore, these all vanish along the projectivization of the distinguished one-dimensional projective subbundle of ℙ(V^{∨}). Divisors D in O_{ℙ(V^{∨})(2)} ⊗ π^{∗}O_{ℙ^{1}}(−1) are fibered over ℙ^{1} in quadrics containing a distinguished line ℓ. Let X = D_{1} ∩ D_{2} for two divisors in this linear series.

Remark 19. Since D_{1} and D_{2} both contain ℓ, we have a constant subfibration:

ℓ × ℙ^{1} ⊂ X
↓ ↙
ℙ^{1}

This provides sections for X → ℙ^{1}. Note here that X is non-minimal, as ℓ can be blown down over the ground field.

Odd part. Suppose that the additional forms have degrees (n, 2) and (n + 1, 2). Then ω_{π} ≃ O_{X} (2n + 4, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 3)^{3} ⊕ O_{ℙ^{1}}(−2n − 4)^{2}

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 17, h(X) = 20n + 34.

Example 20 (n = −1 case). Retain the notation of Example 18. Here we let X = D ∩ Q where Q has bidegree (0, 2). The image of ℙ(V^{∨}) in ℙ^{7} lies on three quadrics, and is in fact a cone over a Segre threefold.

The intersection of ℓ with Q gives two distinguished sections

σ_{1}, σ_{2} : ℙ^{1} → X ,

which may be conjugate over the ground field. We shall study this case in Section 11.

Case 5. Consider

X ⊂ ℙ^{1} × ℙ^{8}

as a complete intersection of four forms L_{1}, L_{2}, L_{3} and L_{4} of bidegree (1, 1) and two additional forms quadratic in the second factor. Let V be defined by the exact sequence

0 → O_{ℙ^{1}}(−1)^{4} →^{·(L_{1},L_{2},L_{3},L_{4})} Γ(O_{ℙ^{8}}(1)) ⊗ O_{ℙ^{1}} → V → 0;

for generic L_{i}, we have

V ≃ O_{ℙ^{1}}(1)^{4} ⊕ O_{ℙ^{1}}.

Even part. Suppose that the additional quadratic forms both have degree (n, 2). Then ω_{π} ≃ O_{X} (2n + 4, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 3)^{4} ⊕ O_{ℙ^{1}}(−2n − 4)

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 16, h(X) = 20n + 32.

Example 21 (n = −1 case). Consider the intersection of the forms L_{1}, L_{2}, L_{3} and L_{4}:

ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{4} ⊕ O_{ℙ^{1}}) ⊂ ℙ(O_{ℙ^{1}}^{9}).

In this case, we have

Sym^{2}(V) ⊗ O_{ℙ^{1}}(−1) = O_{ℙ^{1}}(1)^{10} ⊕ O_{ℙ^{1}}^{4} ⊕ O_{ℙ^{1}}(−1),

and the space of sections has dimension 24. These all vanish along a distinguished section σ : ℙ^{1} → ℙ(V^{∨}), corresponding the projectivization of the trivial summand of V^{∨}; let D_{1} and D_{2} denote two of the divisors in this class. Let X = D_{1} ∩ D_{2}, which is a quartic del Pezzo fibration.

Remark 22. The section σ factors through X. This canonical section is defined over any field of scalars.

Odd part. Suppose that the additional forms have degrees (n, 2) and (n + 1, 2). Then ω_{π} ≃ O_{X} (2n + 5, −1) whence

π_{∗}ω_{π}^{−1} ≃ O_{ℙ^{1}}(−2n − 4)^{4} ⊕ O_{ℙ^{1}}(−2n − 5)

and

deg(π_{∗}ω_{π}^{−1}) = −10n − 21, h(X) = 20n + 42.

We can realize the n = −1 case via constructions similar to Examples 18, 16, 20, and 21.

Example 23. A direct generalization of the approach in Examples 18, 16, 20, and 21 yields a pencil of singular del Pezzo surfaces. The bundle corresponding to the bidegree (−2, 2) forms is

Sym^{2}(V) ⊗ O_{ℙ^{1}}(−2) = O_{ℙ^{1}}^{10} ⊕ O_{ℙ^{1}}(−1)^{4} ⊕ O_{ℙ^{1}}(−2),

parametrizing quadric hypersurfaces singular along the distinguished section of ℙ(V^{∨}). The corresponding divisors D depend on 9 parameters. The bundle corresponding to bidegree (−1, 2) forms is

Sym^{2}(V) ⊗ O_{ℙ^{1}}(−1) = O_{ℙ^{1}}(1)^{10} ⊕ O_{ℙ^{1}}^{4} ⊕ O_{ℙ^{1}}(−1),

parametrizing quadric hypersurfaces containing the distinguished section. The associated divisor D′ depends on 24−1−2 = 21 parameters. Altogether, this construction depends on

56 + 9 + 21 − (3 + 80) = 3

parameters, whereas the expected dimension is two. This reflects the higher cohomology of the bundle Sym^{2}(V)(−2).

5. Sections of height two fibrations

There are no examples of height-two quartic del Pezzo fibrations X → ℙ^{1} with smooth total space and fibers having at worst one ordinary double point. We will address this in a future paper. For the moment, we analyze sections in the context of Example 23.

Projecting from the distinguished section p : ℙ^{1} → X gives a rational map

ℙ(O_{ℙ^{1}}(−1)^{4} ⊕ O_{ℙ^{1}}) −−→ ℙ(O_{ℙ^{1}}(−1)^{4}) ≃ ℙ^{3} × ℙ^{1}.

The image of D under this mapping is a smooth, trivial quadric surface fibration

Q := Q × ℙ^{1} ⊂ ℙ^{3} × ℙ^{1} →^{ϖ} ℙ^{1}.

Let E = E × ℙ^{1} denote the exceptional divisor over p(ℙ^{1}); E ⊂ Q is a hyperplane section.

The equation of D′ takes the form

H(u_{1}, u_{2}, u_{3}, u_{4})u_{0} + sF (u_{1}, u_{2}, u_{3}, u_{4}) + tG(u_{1}, u_{2}, u_{3}, u_{4}) = 0,

where H is linear and F and G are quadratic. Here s and t are homogeneous coordinates on ℙ^{1}, u_{1}, …, u_{4} are homogeneous coordinates on ℙ^{3}, and [u_{0}, u_{1}, …, u_{4}] = [1, 0, …, 0] is the distinguished section. Thus X → Q blows up the curve

Z = {H = sF + tG = 0} ⊂ E = E × ℙ^{1} = {H = 0},

a type (4, 1) curve in E × ℙ^{1} ≃ ℙ^{1} × ℙ^{1}. The projection ϖ_{2} : E → E induces Z ≃ E. In summary, we have

X̃ = Bl_{p(ℙ^{1})}(X) = Bl_{Z}(Q)
↙  ↘
X  Q

and

h^{1}(Ω2 ; X̃) = h^{2}(Ω1 ; X̃) = 0.

We enumerate sections σ : ℙ^{1} → X disjoint from the singular section.

height 0: The points of

Q \ (Q ∩ {H = 0}) ≃ (ℙ^{1} × ℙ^{1}) \ {diagonal}

yield height-zero sections of π : X → ℙ^{1} disjoint from p(ℙ^{1}).

height 1: Fix a line L ⊂ Q from one of the two rulings, which meets E in a single point. The subfibration L×ℙ^{1} ⊂ Q meets Z in a single point z. Curves of bidegree (1, 1) in L×ℙ^{1} containing z form a two-parameter family. The proper transforms of these curves in X are sections disjoint from p(ℙ^{1}). Altogether, we obtain a three parameter space of sections, birational to ℙ^{1} × ℙ^{2}. Taking the choice of ruling into account, we obtain two such components.

Given a generic point x′ ∈ X, we exhibit height-one sections of π containing x′. Regard x′ ∈ Q × p′, p′ = π(x′) and choose the ruling so it contains the projection of x′ onto Q. These are indexed by two copies of ℙ^{1} \ {two points}, one for each ruling of Q. height 2: Fix a hyperplane section M ⊂ Q, meeting E in two points; M × ℙ^{1} meets Z in {z_{1}, z_{2}}. Curves of bidegree (1, 1) in M × ℙ^{1} containing {z_{1}, z_{2}} form a one-parameter family. This construction gives a four parameter space of sections of X → ℙ^{1}, birational to ℙ^{3} × ℙ^{1}.

height 3: Repeat the same argument for curves of bidegree (1, 2) (twisted cubics) in Q.

height 4: Repeat the same argument for curves of bidegree (1, 3) in Q.

Consider nodal rational curves M′ ⊂ Q of bidegree (2, 2), which meet E in four points. Let M denote the normalization of M′; M ×ℙ^{1} meets Z in four points, {z_{1}, z_{2}, z_{3}, z_{4}}. Consider curves of bidegree (1, 1) in M × ℙ^{1} containing these points. Such curves exist only when we have a projective equivalence

(M; z_{1}, …, z_{4}) ∼ (ℙ^{1}, ϖ(z_{1}), ϖ(z_{2}), ϖ(z_{3}), ϖ(z_{4})).

This leads to the following geometric problem:

Let Q be a smooth quadric surface, E ⊂ Q a smooth hyperplane, z_{1}, …, z_{4} ∈ E prescribed points, and w_{1}, …, w_{4} ∈ ℙ^{1} prescribed points. Describe the morphisms

j : ℙ^{1} → Q

of bidegree (2, 2) such that j(w_{i}) = z_{i} for i = 1, …, 4. These are constructed as follows: For any point q ∈ Q, there exists a curve of bidegree (2, 2) in Q with a node at q, containing z_{1}, …, z_{4}, and satisfying the condition on projective equivalence: Indeed, projection from q converts the problem to finding a conic through four given points in ℙ^{2} so that the points have prescribed projective equivalence class in the conic.

height 5: The case of bidegree (1, 4) proceeds just as the case of twisted cubics.

The case of bidegree (2, 3) boils down to the following:

Let Q be a smooth quadric surface, E ⊂ Q a smooth hyperplane, z_{1}, …, z_{5} ∈ E prescribed points, and w_{1}, …, w_{5} ∈ ℙ^{1} prescribed points. Describe the morphisms

j : ℙ^{1} → Q

of bidegree (2, 3) such that j(w_{i}) = z_{i} for i = 1, …, 5.

These are constructed as follows: Fix pairs of pointed stable curves

(E; z_{1}, …, z_{5}), (ℙ^{1}; w_{1}, …, w_{5}), E ≃ ℙ^{1}

and let C denote the stable curve of genus four obtained by gluing z_{i} to w_{i} for i = 1, …, 5. The canonical embedding of C ⊂ ℙ^{3} is contained in a quadric surface

Q′ ≃ ℙ^{1} × ℙ^{1} ⊂ ℙ^{3};

we may assume the components E and ℙ^{1} have types (1, 2) and (2, 1) respectively. For any two general points e_{1}, e_{2} ∈ E, consider the linear series of curves of type (1, 2) with assigned basepoints at e_{1} and e_{2}. This yields a morphism to a quadric surface

Bl_{e_{1},e_{2}}Q′ → Q ⊂ ℙ^{3}.

The image of E is a hyperplane class; the image of ℙ^{1} is of type (2, 3).

Thus for each collection {z_{1}, …, z_{5}} ⊂ E, we obtain a two-parameter space of sections σ : ℙ^{1} → X, corresponding to M′ ⊂ Q of bidegree (2, 3) such that

• M′ ∩ Q = {z_{1}, …, z_{5}};

• (M; z_{1}, …, z_{5}) ≃ (ℙ^{1}; π(z_{1}), …, π(z_{5})).

The two parameters reflect the choice of (e_{1}, e_{2}) above. Hence we get a seven-dimensional rational variety parametrizing these sections.

height 6: We focus on the case of bidegree (3, 3), which leads to the following statment:

Let Q be a smooth quadric surface, E ⊂ Q a smooth hyperplane, z_{1}, …, z_{6} ∈ E prescribed points, and ϕ : E ≃ ℙ^{1} → ℙ^{1} a fixed mapping of degree four. The morphisms

j : ℙ^{1} → Q

of bidegree (3, 3) such that

(ℙ^{1}; ϕ(z_{1}), …, ϕ(z_{6})) ∼ (E; z_{1}, …, z_{6})

are parametrized by a rational variety. The rational curves of bidegree (3, 3) through six fixed points depend on 15 − 4 − 6 = 5 parameters.

This is proved as follows: Let C be a smooth projective curve of genus five that is Brill-Noether general, i.e., not hyperelliptic, trigonal, or a plane quintic. Then the canonical embedding

C ↪ ℙ^{4}

realizes C as a complete intersection of three quadrics. In particular, C admits embeddings into a various quartic del Pezzo surfaces parametrized by ℙ(I_{C}(2)^{∨}) ≃ ℙ^{2}. This remains true for certain stable curves, e.g.,

C = C_{1} ∪_{z_{1},...,z_{6}} C_{2}, C_{1} ≃ C_{2} ≃ ℙ^{1},

where z_{1}, …, z_{6} are chosen arbitrarily on each component.

Fixing (C_{1}; z_{1}, …, z_{6}) and (C_{2}; z_{1}, …, z_{6}), let S denote one of the del Pezzo surfaces arising from this construction. The linear series |C_{1}| induces a birational morphism onto a quadric

S → Q.

The image of C_{1} is Q is a hyperplane section; the image of C_{2} is a bidegree (3, 3) curve with four nodes.

6. Sections of height four fibrations

We summarize spaces of sections of small height for height four fibrations.

Recall the notation of Example 18. The generic fiber X contains a line ℓ. Projecting from this line gives a birational morphism

X → ℙ^{2}

blowing down the five lines incident to ℓ; ℓ is mapped to the plane conic containing the images of the five lines.

We interpret this for X over ℙ^{1}: Fiberwise projection from ℓ × ℙ^{1} induces a rational map

ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{3} ⊕ O_{ℙ^{1}}^{2}) −−→ ℙ(O_{ℙ^{1}}(−1)^{3}) ≃ ℙ^{2} × ℙ^{1},

inducing a birational morphism

β : X → ℙ^{2} × ℙ^{1}.

The image L := β(ℓ × ℙ^{1}) ⊂ ℙ^{2} × ℙ^{1} is a product {conic} × ℙ^{1}. Let ϖ_{j}, j = 1, 2 denote the projections from ℙ^{2} × ℙ^{1}.

We describe the center Z of β: It is a rational curve of bidegree (5, 1) in ℓ × ℙ^{1}, i.e., projection onto the second factor has degree five and eight branch points. It follows that

h^{2}(Ω_{X}^{1} ) = 0, h^{1}(Ω_{X}^{1} ) = 3.

The composition

ℙ^{2} × ℙ^{1} −−→^{β^{−1}} X →^{π_{2}} Y

is given by forms of bidegree (3, 1) on ℙ^{2} × ℙ^{1} vanishing along Z. Its restriction to ℓ × ℙ^{1} ⊂ X has bidegree (2 · 3, 1) − (5, 1) = (1, 1), thus collapses ℓ × ℙ^{1} onto the first factor.

We enumerate sections of π : X → ℙ^{1} of small height h, restricting our attention to sections that dominate X. For example, we do not consider sections contained in ℓ×ℙ^{1}. Furthermore, we subdivide based on how the sections meet ℓ × ℙ^{1} and E, the exceptional divisor of β.

height 0: The constant sections of ℙ^{2} × ℙ^{1} → ℙ^{1} induce a twoparameter space of sections for π : X → ℙ^{1}, disjoint from ℓ × ℙ^{1} and E.

height 1: Consider sections of ϖ_{2} : ℙ^{2}×ℙ^{1} → ℙ^{1} projecting onto a line L ⊂ ℙ^{2}, meeting Z in {z_{1}, z_{2}} with ϖ_{1}(z_{1}, z_{2}) = L ∩ ℓ. These depend on three parameters, are disjoint from ℓ × ℙ^{1}, but meet E twice. These sections are parametrized by a space fibered over Sym^{2}(Z) ≃ ℙ^{2}, with generic fiber ℙ^{1}.

height 2: We have sections of ϖ_{2} projecting onto a line L, meeting Z at one point. The resulting sections of X → ℙ^{1} depend on four parameters, and meet both E and ℓ × ℙ^{1} once.

There are also sections with π_{1}(σ(ℙ^{1})) = M a plane conic, where the corresponding curve in M × ℙ^{1} meets Z in four points {z_{1}, z_{2}, z_{3}, z_{4}} with π_{1}(z_{1}, z_{2}, z_{3}, z_{4}) = M ×ℓ. These are birational to Sym^{4}(Z); indeed, there exists a unique plane conic M ⊃ ϖ_{1}(z_{1}), …, ϖ_{1}(z_{4}) such that the projective invariant of these four points in M equals the projective invariant of the points π(z_{1}), …, π(z_{4}) in ℙ^{1}.

height 3: Sections of ϖ_{2} projecting onto a line and disjoint from Z. These are parametrized by an open subset (ℙ^{2})^{∨} × ℙ^{3}; the generic fiber over a point of (ℙ^{2})^{∨} is isomorphic to ℙ^{3} \ {two hyperplanes}. The resulting sections of π are disjoint from E and meet ℓ × ℙ^{1} twice.

There are sections of ϖ_{2} projecting onto a conic, meeting Z in three points. We have a unique such section over each conic in ℙ^{2}, so the space of these sections is birational to ℙ^{5}. The corresponding sections of π meet E three times and ℓ × ℙ^{1} one.

Sections of ϖ_{2} projecting onto a nodal cubic curve M′ with normalization M; the corresponding curve in M × ℙ^{1} meets Z in six points {z_{1}, …, z_{6}}. We have the constraint that

(M; z_{1}, …, z_{6}) ∼ (ℙ^{1}; π(z_{1}), …, π(z_{6})),

i.e., these are projectively equivalent. Projection from the node of M′ induces a degree-two cover

ℓ → M

and ℓ ≃ Z via ϖ_{1}. This translates into the following problem

Fix a degree five morphism ϕ : ℙ^{1} → ℙ^{1}. Characterize sets of points

(z_{1}, …, z_{6}) ∈ (ℙ^{1})^{6}

such that there exists some degree-two morphism ψ : ℙ^{1} → ℙ^{1} with

(ℙ^{1}; ϕ(z_{1}), …, ϕ(z_{6})) = (ℙ^{1}; ψ(z_{1}), …, ψ(z_{6})).

Note that ψ coincides with projection from the node of M′ and determines this node uniquely.

This example can be approached using the Brill-Noether analysis employed in Section 5. Consider pairs of 6-pointed stable curves of genus zero

(C_{1}; z_{1}, z_{2}, …, z_{6}), (C_{2}; z_{1}, z_{2}, …, z_{6}),

and let C denote the genus-five stable curve obtained by gluing these together. If C is immersed in ℙ^{2} as the union of a conic and a nodal cubic

M′ ∪ ℓ ⊂ ℙ^{2}

then the canonical curve j : C ↪ ℙ^{4} lies on a cubic ruled surface, i.e., the blow up of ℙ^{2} at the node. This is equivalent to C being trigonal, with the g_{3}^{1} given by the rulings. Trigonal curves are codimension one in the moduli space of curves of genus five.

The projection of the trigonal curves C as above to M_{0,6} × M_{0,6} is finite onto its image, by a direct computation with MAPLE. This image is unirational as it is dominated by ℙ^{5} ×ℙ^{6}—the choice of a plane conic and a plane cubic with a node at a prescribed point.

7. Sections of height six fibrations

We describe sections of small height for this class of fibrations. Recall the notation leading into Remark 17. In particular, we have

ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{2} ⊕ O_{ℙ^{1}}^{3}) →^{π_{2}} ℙ^{6},

where the image is quadric hypersurface. It contains a distinguished plane Π = π_{2}(ℙ(O_{ℙ^{1}}^{3})). The divisor D ⊂ ℙ(V^{∨}) has image π_{2}(D) isomorphic to a cone over the Segre threefold ℙ^{1} × ℙ^{2} ⊂ ℙ^{5}. The plane Π is the cone over ℙ^{1} × {p} for a point p ∈ ℙ^{2}.

Since Y is the intersection of π_{2}(D) with a quadric hypersurface Q, we may interpret it as a double cover of ℙ^{1} × ℙ^{2} branched over a divisor of bidegree (2, 2). In particular, Y is a quadric surface bundle over ℙ^{1} with six degenerate fibers, and a conic bundle over ℙ^{2}, with discriminant a plane quartic Z ⊂ ℙ^{2} and associated étale double cover γ : Z̃ → Z. Thus the intermediate Jacobian

IJ(Y) = Prym(Z̃ → Z) = J(C),

where C is the genus two curve branched over the discriminant of Y → ℙ^{1}. Thus in particular, we have

h^{2}(Ω_{X}^{1} ) = 2, h^{1}(Ω_{X}^{1} ) = 3,

which reflects the fact that the fibration π : X → ℙ^{1} has relative Picard rank two.

Proposition 24. There exists a space of sections of π : X → ℙ^{1} mapping birationally onto IJ(X) ≃ Prym(Z̃ → Z).

Proof. The morphism π_{2} : X → Y is the blow up of Y along the curve T = Π ∩ Q, which coincides with the fiber of Y → ℙ^{2} over p. The fibration

π = π_{1} : X → ℙ^{1}

is induced by the projection from p

ϖ_{p} : ℙ^{2} −−→ ℙ^{1}.

Let ℙ̌^{2} denote the projective space dual to the ℙ^{2} arising as a factor of the Segre threefold. For each L ∈ ℙ̌^{2}, we have a conic bundle

ϕ_{L} : Y ×_{ℙ^{2}} L → L,

degenerate over L∩Z. This conic bundle has eight (−1)-curves E_{1}, …, E_{8} that induce sections {s_{1}, …, s_{8}} of ϕ_{L}; we obtain a generically finite morphism

E → ℙ̌^{2},

where E is a smooth projective surface compactifying the space parametrizing E_{1}, …, E_{8} as L varies. In particular, there is an open set E^{∘} ⊂ E and a correspondence

\begin{matrix}S & → & Y \\ ↓ \\ E^{∘} \end{matrix}

where the vertical arrow is a ℙ^{1}-bundle and the horizontal arrow is induced by the inclusions E_{1} ∪ … ∪ E_{8} ⊂ Y ×_{ℙ^{2}} L ⊂ Y.

Now each E_{j} meets each of the four degenerate fibers of ϕ_{L} in one component; these components are indexed by z_{1}, z_{2}, z_{3}, z_{4} ∈ Z̃. Note that γ(z_{1}, z_{2}, z_{3}, z_{4}) = L ∩ Z ∈ |K_{Z}|. Thus we obtain maps

E −−→ Pic^{4}(Z̃) →^{γ_{∗}} Pic^{4}(Z),

where the image of E lies in the fiber over K_{Z}. Thus there is an induced mapping

E −−→ Prym(Z̃ → Z),

which is clearly birational.

We claim that the fibers of S → E yield sections of π : X → ℙ^{1}, i.e., E parametrizes a space of sections birational to IJ(X). Indeed, fix [E_{j}] ∈ E over L ⊂ ℙ^{2} and a fiber X_{r} = π^{−1}(r) for r ∈ ℙ^{1}. The point r corresponds to a line

ℓ_{r} := ϖ_{p}^{−1}(r) ⊂ ℙ^{2}

passing through p. There is a unique intersection point p′ = L ∩ ℓ_{r}, and the fiber Y ×_{ℙ^{2}} {p′} meets E_{j} in precisely one point. □

8. Sections of height eight fibrations

We return to the example discussed in Remark 15: Let Y ⊂ ℙ^{5} denote a complete intersection of two quadrics, E ⊂ Y a smooth codimension-two linear section, and X = Bl_{E}(Y). The pencil of linear sections containing E induces a del Pezzo fibration

π : X → ℙ^{1}.

We examine sections of π of small height.

We review the classical constructions for the lines on Y, following Miles Reid’s thesis [Rei72] (see also [Tju75] and [Cas04] for higherdegree curves). The discriminant divisor restricted to the pencil of quadrics yields six points on ℙ^{1}, thus determining a genus two curve C. Note that the space of maximal isotropic subspaces on quadrics in the pencil is a ℙ^{3}-bundle over C; these correspond to the conics on Y. Projecting from a line ℓ ⊂ Y, we obtain a morphism

Bl_{ℓ}(Y) → ℙ^{3},

blowing up C, realized as a quintic curve in ℙ^{3}. The space of all lines

F_{1}(Y) ≃ J(C),

which in turn is isomorphic to the intermediate Jacobian IJ(Y). Thus we conclude

IJ(X) = J(C) × E.

Proposition 25. π has a space of sections birational to a ℙ^{1}-bundle over a principal homogeneous space for IJ(X).

Before proving this, we point out how sections of π may be interpreted in terms of the geometry of Y:

height 0: Lines on Y not incident to E correspond to sections of π, parametrized by IJ(Y).

height 1: Conics on Y incident to E correspond to a three-dimensional space of sections. This a priori has a chance to dominate IJ(X). However, the conics in Y fail to dominate IJ(Y), so the space we’re considering cannot possibly dominate IJ(X).

height 2: Twisted cubics on Y incident to E in two points also yield sections; these are relevant for the proof of Proposition 25.

Proof. The space of twisted cubics in Y is birational to a Grassmannbundle over the lines in Y: given ℓ ⊂ Y, each subspace

ℓ ⊂ ℙ^{3} ⊂ ℙ^{5}

intersects Y in the union of ℓ and a twisted cubic R. Given e_{1}, e_{2} ∈ E, consider

span(ℓ, e_{1}, e_{2}) ∩ Y = ℓ ∪ R,

which necessarily meets E in e_{1} and e_{2}. The curves produced in this way are parametrized by

J(C) × Sym^{2}(E),

a ℙ^{1}-bundle over J(C) × E.

These one-parameter families of sections sweep out a distinguished rational curve in X, the generic fiber of X → ℙ^{1}. Fix a pair (Q, R) consisting of • a quadric surface E ⊂ Q ⊂ span(Q);

• a ruling R ∈ Pic(Q) such that O_{Q}(R)|E = O_{E}(e_{1} + e_{2}). There is a natural bijection between such pairs and elements of Pic^{2}(E). Consider the singular quadric C_{ℓ}(Q) ⊂ ℙ^{5} that is a cone over Q with vertex ℓ. The intersection of C_{ℓ}(Q) with a fiber X_{t} of X → ℙ^{1} consists of E plus a hyperplane section singular at ℓ ∩ X_{t}. □

9. Sections of height ten fibrations

We refer to Case 1 of Section 4, specifically, the n = 0 instance of the ‘Odd part’. Let Q ⊂ ℙ^{4} be a smooth quadric threefold. Consider a pencil of quadrics on Q with smooth base locus C, a canonical curve of genus 5. Thus we have a del Pezzo fibration

π : X := Bl_{C}(Q) → ℙ^{1}

with fibers having at most one ordinary double point.

Proposition 26. π has a space of sections parametrized by a principal homogeneous space over the Jacobian J(C) ≃ IJ(X).

As a prelude, we describe sections of small height as curves on Q incident to C:

height −1: These are fibers of the exceptional divisor of X → Q, parametrized by points of C.

height 0: These are lines on Q incident to C at one point, a ℙ^{1}-bundle over C.

height 1: Conics on Q three-secant to C, birationally parametrized by Sym^{3}(C). indeed, take the plane spanned by three points on C and intersect this with Q.

height 2: These are twisted cubics R ⊂ Q meeting C in five points, which are birational to a ℙ^{1}-bundle over Sym^{3}(C). Given such a twisted cubic, let H ⊂ ℙ^{4} be the hypersurface it spans; it contains a quadric surface (Q∩H) and eight distinguished points (C ∩H) on that surface. Five of these points are on R; three are not. Thus we have a mapping

{ twisted cubics meeting C five times } −−→ Sym^{3}(C).

The fiber over p + q + r ∈ Sym^{3}(C) consists of pairs

{(H, R) : H ⊃ {p, q, r}, (C ∩ H) \ {p, q, r} ⊂ R ⊂ Q ∩ H,

which is a double cover over ℙ^{1} via projection onto the first factor. Indeed, there are two twisted cubics passing through five generic points on a quadric surface. However, given a pencil of three-planes containing a plane in ℙ^{4}, two elements will be tangent to a quadric hypersurface Q. Thus the double cover over ℙ^{1} is branched in two points, so the fibers of the mapping are rational curves.

height 3: These are quartic rational curves R ⊂ Q meeting C in seven points—the sections relevant for the proof of Proposition 26.

Proof. We interpret this via an enumerative problem:

Fix D ∈ Pic^{7}(C); then there exists a unique twisted quartic R ∈ Q such that R ∩ C ∈ |D|.

Consider the linear series |D|. Through each collection of seven points Σ = {c_{1}, …, c_{7}} in linear general position there passes a unique twisted quartic R(Σ). We are interested in the locus

Y = ∪_{Σ∈|D|}R(Σ) ⊂ ℙ^{4}.

The linear system |D| realizes C as a plane septic curve with 10 double points x_{1}, …, x_{10}. These depend on a total of 36 − 10 − 9 = 17 parameters, and x_{1}, …, x_{10} can be in general position. Consider the image of ℙ^{2} under the linear series of quartics through x_{1}, …, x_{10}, i.e., the adjoint linear series. The resulting surface Y has degree 6 in ℙ^{4}.

The quadric hypersurface Q pulls back to a plane curve of degree eight double at x_{1}, …, x_{10}. Of course, this contains C and the residual curve is a line in ℙ^{2}. Thus we find

Y ∩ Q = C ∪ R

where R is a twisted quartic in ℙ^{4}.

Now we are essentially done: There exists a space of twisted quartic curves

\begin{matrix}R & → & X \\ ↓ \\ W \end{matrix}

whose fibers each meet C seven times. The induced mapping

μ : W → Pic^{7}(C)

is birational. However, Pic^{7}(C) is a principal homogeneous space over the Jacobian of C. □

Question 27. Where else does this periodic behavior occur for minimal quartic del Pezzo surfaces over F = k(t)? Can this be analyzed via moduli of vector bundles and Serre’s construction?

10. Sections of height twelve fibrations

Recall the notation of Example 21. We have two projections

X ⊂ ℙ(V^{∨}) = ℙ(O_{ℙ^{1}}(−1)^{4} ⊕ O_{ℙ^{1}}) →^{π_{2}} ℙ^{8}
↓π
ℙ^{1}

and let Y denote the image of X under π_{2}. The canonical section σ(ℙ^{1}) is contracted under π_{2} to a double point y ∈ Y. The image is a nodal Fano threefold of genus seven and degree twelve. See [Muk95] for concrete descriptions of smooth Fano varieties of this type.

There is an alternate approach to X: Project X → ℙ^{1} from the distinguished section σ(ℙ^{1})

X̃ = Bl_{σ(ℙ^{1})}(X) ⊂ ℙ(O_{ℙ^{1}}(−1)^{4}) ≃ ℙ^{1} × ℙ^{3};

let E denote the exceptional divisor. The resulting fibration

π̃ : X̃ → ℙ^{1}

is a cubic surface fibration containing a constant line E. Now projection onto the second factor

π̃_{2} : X̃ → ℙ^{3}

is birational and collapses E onto a line E ⊂ ℙ^{3}. The center of π̃_{2} is the union of E and a curve C, of genus seven, admitting a line bundle M with h^{0}(M, C) = 4, and deg(M) = 8, i.e., a g_{8}^{3}; the divisor K_{C} − M is a g_{4}^{1}. Note that E and C meet in four points r_{1}, r_{2}, r_{3}, and r_{4}; the fibers of π̃_{2} are mapped onto the pencil of cubic surfaces cutting out E ∪ C. The birational mapping obtained by composing the inverse of π̃_{2}, the blow up of X, and π_{2}

ℙ^{3} −−→ Y ⊂ ℙ^{8}

is induced by the linear series of quartics vanishing along C ∪ E.

Note that the intermediate Jacobian IJ(X) ≃ J(C). This also has a Prym interpretation: Projecting from E fiberwise allows us to realize X̃ as a conic bundle

X̃ → ℙ^{1} × ℙ^{1},

with discriminant a genus-eight curve D of bidegree (3, 5). If D̃ → D is the cover arising from the conic bundle then

IJ(X) ≃ Prym(D̃ → D).

This is an instance of the tetragonal construction of Donagi and Recillas [Don81]. Proposition 28. π admits a space of sections birational to the Jacobian J(C) ≃ IJ(X).

Before proving this, we characterize sections of π of small degree using the geometry sketched above:

height −2: the canonical section σ(ℙ^{1});

height −1: the sections arising from exceptional fibers over points of C, which depend on one parameter;

height 0: the sections arising from secant lines to C ⊂ ℙ^{3}, which are parametrized by Sym^{2}(C);

height 1: the sections arising from conics five-secant to C, which are parametrized by Sym^{3}(C);

height 2: the sections arising from twisted cubics meeting C in eight points, which are parametrized by Sym^{4}(C);

height 3: the sections arising from eleven-secant quartic rational curves, which are parametrized by Sym^{5}(C);

height 4: the sections arising from quintic rational curves 14-secant to C, which are parametrized by Sym^{6}(C);

height 5: the sections arising from sextic rational curves 17-secant to C, which are parametrized by Sym^{7}(C).

We first address sections of height 1, i.e., the conic case: each triple of points on C spans a plane, whose residual intersection with C consists of five points that uniquely determine a five-section conic.

We do sections of height 2: Given such a section R meeting C in p_{1}, …, p_{8}, there is a three-dimensional vector space of quadrics cutting out R, which induces a g_{8}^{2} on C. Note that the variety of g_{8}^{2}’s on a genus seven curve is four-dimensional, reflecting the fact that a g_{8}^{2} is of the form K_{C} − c_{1} − c_{2} − c_{3} − c_{4} for c_{1}, c_{2}, c_{3}, c_{4} ∈ C. This gives a morphism from the space of sections to Sym^{4}(C).

Conversely, suppose we are given generic points c_{1}, c_{2}, c_{3}, c_{4} ∈ C. Choose a twisted cubic R′ incident to E at two points and containing c_{1}, …, c_{4}. (The curve R′ is a tri-section of our original del Pezzo fibration.) The union R′ ∪ E ∪ C has degree 12 and arithmetic genus 15, thus

dim I_{R′∪E∪C}(4) = 35 − (48 + 1 − 15) = 1

and the curve R′ ∪ E ∪ C sits on a quartic surface S. Note that this generically has Picard lattice:

h C E R′
h 4 8 1 3
C 8 12 4 4
E 1 4 −2 2
R′ 3 4 2 −2

Let F be the residual to R′ ∪ E in a complete intersection of a quadric surface and S; this is an elliptic quartic curve. Consider the union C ∪ F ∪ E; we have

h^{0}(I_{C∪F ∪E}(4)) ≥ 35 − h^{0}(O_{C∪F ∪E}(4))

= 35 − (4 deg(C ∪ F ∪ E) + 1 − g(C ∪ F ∪ E))

= 35 − (52 + 1 − 20) = 2.

Let R be residual to C ∪F ∪E in this complete intersection. Note that

C · R = 8, R · R = −2,

which gives the desired section.

We turn to sections of height 3: First we construct the morphism from the space of sections to Sym^{5}(C). Suppose that R is a rational quartic curve in ℙ^{3} meeting C ⊂ ℙ^{3} in eleven points. Choose a K3 surface S containing C and R, with Picard lattice:

h C R
h 4 8 4
C 8 12 11
R 4 11 −2

These exist as

dim I_{C∪R}(4) ≥ 35 − (48 + 1 − 17) = 3.

Now R is also contained in a quadric surface Q, generally unique. Let R′ denote the residual to R in S ∩ Q, so that

h · R′ = 4, R′ · R′ = −2, C · R′ = 5.

The intersection of C with R′ gives five points on C.

We refer the reader to [HT12a, Sect. 8] for the inverse construction.

Now for the case of sections of height 4: Given six points c_{1}, …, c_{6} ∈ C, let R′ denote the unique twisted cubic in ℙ^{3} through these points. Consider the pencil of quartic surfaces containing the union

C ∪_{c_{1},...,c_{6}} R′,

each of which has the lattice polarization:

h C R′
h 4 8 3
C 8 12 6
R′ 3 6 −2

Consider the residual curve R to C ∪ R′ in this complete intersection; it has genus zero, degree five, and meets C in fourteen points. Thus R corresponds to a section of π; these are parametrized by elements of Sym^{6}(C), which may be interpreted as the theta divisor of J(C).

Finally, we turn to sections of height 5, corresponding to sextic rational curves, and the proof of Proposition 28:

Proof. We construct the morphism from the space of sections to Sym^{7}(C). Suppose that R is 17-secant to C. The union C ∪ R has genus 23 and degree 14 so by Riemann-Roch

h^{0}(O_{C∪R}(4)) ≥ 56 + 1 − 23 = 34

and thus is contained in a quartic surface S. This has Picard group:

h C R
h 4 8 6
C 8 12 17
R 6 17 −2

Consider the elliptic fibration C − h; it has R′ = 3h − R as a section

(C − h) · R′ = 1, C · R′ = 7.

Geometrically, this arises from R via residuation: A rational sextic space curve R satisfies

h^{0}(I_{R}(3)) ≥ 20 − (18 + 1) = 1

and thus lies in a cubic surface T. Then R′ is the residual to R in T ∩S. The intersection of R′ with C consists of seven points, which gives us our point in Sym^{7}(C).

This construction is invertible by the following result, proven in [HT12a]:

Let C ⊂ ℙ^{3} denote a generic tetragonal curve of genus 7 as above, and c_{1}, …, c_{7} ∈ C generic points. Then there exists a unique sextic rational curve R′ ⊂ ℙ^{3} and quartic K3 surface S ⊂ ℙ^{3} such that

R′ ∩ C = {c_{1}, …, c_{7}}

and

C ∪ R′ ⊂ S.
□

Remark 29. There is a straightforward construction of a space of sections parametrized by a rationally connected fibration over J(C). Fix c_{1}, …, c_{7} ∈ C and let R′′ denote a rational quartic curve containing these points. Note that the possible such R′′ correspond to ruled quadric surfaces through c_{1}, …, c_{7}, a double cover of ℙ(I_{c_{1},...,c_{7}}(2)) branched over a quartic plane curve, i.e., a degree two del Pezzo surface. These curves yield five-sections of π : X → ℙ^{1}. Using the argument for Brumer’s theorem [Bru78, Lee], for each such odd-degree multisection there exists a space of sections, obtained via a sequence of residuations, and parametrized by a rationally connected variety.

We summarize the evidence for the existence of sections over generic fibrations of small height collected so far.

height Existence of sections? Cross reference
0  always  Remark 13
2  –  (see Example 23)
4  always  Remark 19
6  always  Remark 17
8  always  Remark 15
10  always  Remark 14 and §9
12  always  Remark 22

11. Fibrations of height fourteen to twenty

Let π : X → ℙ^{1} be a generic quartic del Pezzo fibration of height 14 + 2m, m = 0, 1, 2, 3. This corresponds to the following cases of Section 4:

• height 14: Example 20 of Case 4;

• height 16: the n = 0 instance of the even part of Case 3;

• height 18: the n = 0 instance of the odd part of Case 2;

• height 20: the n = 0 instance of the odd part of Case 1. Then we have

X ⊂ ℙ^{1} × ℙ^{7−m} →^{π_{2}} ℙ^{7−m}
π ↓
ℙ^{1}

and we write Y = π_{2}(X). We have:

• Y is anticanonically embedded in ℙ^{7−m} of degree 10 − 2m;

• π_{2} : X → Y is small, generically contracting 2^{m+1} sections of π to nodes of Y.

Indeed, the anticanonical divisor of X (as computed via adjunction) coincides with π_{2}^{∗}O_{ℙ^{7−m}}(1). A direct computation with Bezout’s theorem shows that the number of sections contracted by π_{2} is as claimed. (Over non-closed fields, there is no reason for these to be defined over the ground field.)

In the following sections, we make the geometry even more explicit.

Height fourteen. Recall the notation from Example 20. The two distinguished sections are denoted σ_{1}, σ_{2} : ℙ^{1} → X. Projection onto the second factor gives

Y ⊂ π_{2}(D) ⊂ π_{2}(ℙ(V^{∨}))
||  ||
W  Σ.

Note that Σ is a cone over a Segre threefold ℙ^{1}×ℙ^{2} ⊂ ℙ^{5}, with vertex ℓ, a line. The equations of Σ are given by the 2 × 2 minors of the matrix:

(\begin{matrix}u & v & w \\ x & y & z \end{matrix});

we express

ℓ = {u = v = w = x = y = z = 0} ⊂ Σ.

We may interpret W as follows: Fix a fiber {pt.} × ℙ^{2} in the Segre threefold, yielding a subspace

Λ ≃ ℙ^{4} ⊂ Σ, Λ ⊃ ℓ.

Write

Λ = {u = v = w = 0} ⊂ Σ

and

Q′ = {q_{1}(u, v, w)+ul_{1}(x, y, z, s, t)+vl_{2}(x, y, z, s, t)+wl_{3}(x, y, z, s, t) = 0},

where q_{1} is quadratic and the l_{i} are linear. Note that Q′ is necessarily singular along some line ℓ′ ⊂ Λ, and thus has rank at most six; in suitable coordinates, we have

Q′ = {um_{1} + vm_{2} + wm_{3} = 0},

where the m_{i} are linear. Let W denote the residual to Λ in Q′ ∩ Σ, which is contained in the additional quadric hypersurface

Q′′ = {xm_{1} + ym_{2} + zm_{3} = 0}.

Suppose that Λ′ ≃ ℙ^{4} is a cone over a generic fiber {pt.′} × ℙ^{2} in the Segre threefold; then Q′ and Q′′ cut out the same quadric hypersurface on Λ′, i.e.,

Q′ ∩ Λ′ = Q′′ ∩ Λ′.

We claim that W is a linear section of the Grassmannian Gr(2, 5) ⊂ ℙ^{9}. Indeed, let ξ_{ij}, 1 ≤ i < j ≤ 5 denote the Plücker coordinates on the Grassmannian and consider the linear section ξ_{14} = 0. The Plücker relations specialize to the 2 × 2 minors of

(\begin{matrix}ξ_{12} & ξ_{13} & −ξ_{15} \\ ξ_{24} & ξ_{34} & ξ_{45} \end{matrix})

as well as

ξ_{12}ξ_{35} − ξ_{13}ξ_{25} + ξ_{15}ξ_{23} = 0 − ξ_{23}ξ_{45} + ξ_{24}ξ_{35} − ξ_{25}ξ_{34} = 0.

Setting

u = ξ_{12}, v = ξ_{13}, w = −ξ_{15}, x = ξ_{24}, y = ξ_{34}, z = ξ_{45}

and m_{1} = ξ_{35}, m_{2} = −ξ_{25}, and m_{3} = −ξ_{23}, we obtain the equations for W given above.

In particular, W has degree five. Furthermore, a direct calculation shows that W has ordinary double points along ℓ. This reflects the fact that the birational morphism π_{2} : D → W collapses ℓ × ℙ^{1} to ℓ, and any small contraction of a smooth variety is singular.

The variety Y is the intersection of W with a generic quadric Q, and thus has degree ten and two ordinary singularities

{y_{1}, y_{2}} = ℓ ∩ Q = {π_{2}(σ_{1}(ℙ^{1})), π_{2}(σ_{2}(ℙ^{1}))}.

The dualizing sheaf ω_{Y} = O_{Y}(−1), so Y is a Fano variety with ordinary double points, and thus a degeneration of a smooth Fano variety of degree ten and genus six [Nam97]. Degenerations with one node are studied in [DIM11].

We describe the quartic del Pezzo fibration in terms of the geometry of Y ⊂ ℙ^{7}. The fibers are obtained by intersecting the subspaces Λ′ specified above (associated with the fibers of ℙ^{1} × ℙ^{2} → ℙ^{1}) with Q and Q′; note that Q′ ≡ Q′′ (mod Λ′).

Height sixteen. In this case Y ⊂ ℙ^{6} is a complete intersection of three quadrics with four nodes. These have expected dimension

dim Gr(3, Γ(O_{ℙ^{6}}(2))) − dim PGL_{7} − 4 = 23,

which is equal to the expected dimension of the space of quartic del Pezzo fibrations of height sixteen.

Height eighteen. Here Y ⊂ ℙ^{5} is a complete intersection of a quadric and cubic with eight nodes. These have expected dimension 26, which is equal to the expected dimension of the space of quartic del Pezzo fibrations of height eighteen.

Height twenty. Here Y ⊂ ℙ^{4} is a quartic threefold with sixteen nodes; however, these have expected dimension 30, greater than the expected dimension (\frac{3}{2}h(X)−1 = 29) of the quartic del Pezzo fibrations of height twenty.

Consider the Grassmannian Gr(2, Γ(O_{ℙ^{4}}(2))) parametrizing quartic del Pezzo surfaces. A generic conic curve in this Grassmannian corresponds to the rulings of a nonsingular quadric surface in ℙ(Γ(O_{ℙ^{4}}(2))). These can be expressed in equations as

P_{1}s + Q_{1}t = P_{2}s + Q_{2}t = 0

where P_{1}, Q_{1}, P_{2}, Q_{2} ∈ Γ(O_{ℙ^{4}}(2)). In other words

X ⊂ ℙ^{1} × ℙ^{4}

as a complete intersection of two hypersurfaces of bidegree (1, 2). These depend on

2(2 × 15 − 2) − (3 + 24) = 29

parameters. Let σ_{1}, …, σ_{16} : ℙ^{1} → X denote the constant sections, corresponding to the solutions

{y_{1}, …, y_{16}} = {P_{1} = Q_{1} = P_{2} = Q_{2} = 0}.

Projection onto the second factor gives the quartic hypersurface

Y = {P_{1}Q_{2} − Q_{1}P_{2} = 0} ⊂ ℙ^{4},

with nodes at y_{1}, …, y_{16}.

There is a second del Pezzo fibration

Y −−→ ℙ^{1}

constructed as follows. Fix a fiber X_{t} = π^{−1}(t) and its image X_{t} ⊂ Y. The linear series of quadrics vanishing at X_{t} gives a mapping π′, whose fibers are also del Pezzo surfaces of degree four. This is a morphism where X_{t} ⊂ Y is Cartier, i.e., away from y_{1}, …, y_{16}. There exists a small resolution X′ → Y such that the induced

π′ : X′ → ℙ^{1}

is a morphism. We obtain X′ from X by flopping each of σ_{1}, …, σ_{16}. Finally, let Ỹ denote the blow up of Y at each of the y_{i}, which coincides with the closure of the graph of the birational mapping X −−→ X′; the induced morphism

(π, π′) : Ỹ → ℙ^{1} × ℙ^{1}

has genus five canonical curves as fibers.

Projecting from one of the σ_{j}, we obtain a conic bundle

W → ℙ^{1} × ℙ^{1}

branched along a curve D of bidegree (5, 5), of genus 16. The Prym variety therefore has dimension 15; see also [Che06], where the nonrationality of a generic X is established.

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Department of Mathematics, Rice University, MS 136, Houston, Texas 77251-1892, USA

  E-mail address: hassett@rice.edu

Courant Institute, New York University, New York, NY 10012, USA

  E-mail address: tschinkel@cims.nyu.edu

Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA