1 Lecture 4: sheaves and singularities on fibered powers

1.1 Theorem

Let X → B be a family of pos. dim. varieties of gen. type over an irred. base variety B. Then for large enough n, there exixts a positive dimensional variety of general type W_{n} and a dominant rational map X_{B}^{n}/G −−→ W.

1.2 Theorem (CHM)

Let g > 1 be an integer, and fix a subvariety X_{0} ⊂ M_{g}. Let X_{n} ⊂ M_{g,n} be the inverse image. Then for large enough n, the variety X_{n} is of general type.

1.3 Theorem

Let X → B be a family of varieties of general type over an irredicible base variety B. Assume X → B is of maximal variation. Let G ⊂ Aut(X → B). Then for large enough n, the variety X_{B}^{n}/G is of general type.

Claim: 1.3 implies 1.2.

1.4 Proposition

Let X_{0} ⊂ \overline{M}_{g} be a closed integral subscheme, and let X_{1} ⊂ \overline{M}_{g,1} be the reduced scheme underlying its inverse image. Then there exists a projective, normal integral scheme B, and a family C → B of curves of genus g, satisfying the following properties:

1. The natural moduli morphisms B → \overline{M}_{g} and C → \overline{M}_{g,1} are finite, with images X_{0} and X_{1}, respectively.

2. Denote G = Aut_{\overline{M}_{g}}(C → B). Then the

two morphisms B/G → \overline{M}_{g} and C/G → \overline{M}_{g,1} are birational onto their images.

Moreover, in this situation consider the moduli map C_{B}^{n} −−→ \overline{M}_{g,n}. There is a diagonal action of G on C_{B}^{n}, and C_{B}^{n}/G −−→ \overline{M}_{g,n} is again birational onto the image.

Proof:

1. Hilb_{g} → \overline{M}_{g}

2. take étale slices ^{families do not glue)} 3. take Isom., normalize in Galois closure

4. families glue

isom finite

Situation: a. the function field extension K(W_{0}) ⊂ K(B) is Galois. Denote the Galois group G_{0}. Write η for the generic point of B. b. the geometric points of the finite group H = Aut_{η}C_{η} are all rational over K(B) c. for any g ∈ G_{0} there is a B-isomorphism

(C → B, Θ) ⟶^{∼} g^{∗}(C → B, Θ).

Now consider the set

G = \{(g, h) | \begin{matrix}g ∈ G_{0}, \\ h : (C → B) ⟶^{∼} g^{∗}(C → B) \end{matrix} \} .

Note that for any (g, h) ∈ G and g′ ∈ G_{0} we can define

h^{{g′}} : g′^{∗}(C → B) ⟶^{∼} g′^{∗}g^{∗}(C → B)}.

We can now define

(g, h)(g′, h′) = (gg′, h^{{g′}}h′). Exercise: check that this is a group! It is now immediate to verify condition (2) This “works in higher dimension” ^{no moduli space...} Need to show that something is of general type so need compactification, desingularization, group actions, differential forms.

1.5 Theorem (Kollár)

Arakelov,Viehweg, Kawamata

1.6 Lemma (CHM)

Let f : X → B be a dominant morphism of smooth complex projective varieties. Assume the generic fiber is of general type and the morphism is of maximal variation. Then ω_{f} := ω_{X} ⊗ f^{∗}ω_{B}^{−1} is big on X. f : X → B flat morphism of proj. varieties. A big invertible sheaf on X, L any invertible sheaf on B. p_{i} : X_{B}^{k} → X projections, p_{0} : X_{B}^{k} → B.

Then for large k,

p_{0}^{∗}L ⊗ ( ⊗_{i=1}^{k} p_{i}^{∗}A)

big on X_{B}^{k}. ^{proof}

Immediate deduction of FPT

is false

1. invariants singular 2. powers singular

2 Rational Gorenstein Singularities

canonical

Q: is there a nice class of singularities, s.t.

1. ω_{X} makes sense, and

2. ω^{n} pulls back to desingularization ??

2.1 Dualizing complex and Gorenstein singularities

ω_{ℙ^{N}}^{•} is represented by the complex of sheaves with all stages = 0 apart from ω^{(−N)} = ω_{ℙ^{N}} If X ⊂ ℙ^{N} take RHom(O_{X}, ω_{ℙ^{N}}^{•})

well defined locally as. exercise divisor

X is Cohen-Macaulay ⇔

ω_{X}^{(i)} = 0 for i ≠ − dim(X). Denote ω_{X}^{(− dim(X))} by ω_{X}. If X normal, denote ι : X_{sm} → X, then ω_{X} = ι_{∗}ω_{X_{sm}} X Gorenstein if Cohen-Macaulay and ω_{X} invertible. ^{examples}

2.2 Rational singularities

Let X be a variety and f : X′ → X a desingularization. We say that X has rational singularities if Rf_{∗}O_{X′} = O_{X}.

expand

smooth, toric,...

By duality, this is the same as saying that Rf_{∗}ω_{X′}^{•} = ω_{X}^{•}.

∃ morphism anyway

In particular: X is normal, Cohen-Macaulay. Using G-R vanishing, we have:

2.3 Proposition

Assume X Gorenstein, X′ → X a desingularization, and f_{∗}ω_{X′} = ω_{X}. Then X has rational singularities. Note that it suffices if X′ has rational Gorenstein singularities.

2.4 Observation

If X has rational Gorenstein singularities, and X′ → X a desingularization, then f_{∗}ω_{X′}^{n} = ω_{X}^{n} for all n ≥ 0.

V with rational Gorenstein singularities, G ⊂ Aut V a finite group, Σ ⊂ V fixed points of nontrivial elements of G, q : V → W = V/G the quotient morphism, r : W′ → W a desingularization.

Assume s ∈ H^{0}(V, ω_{V}^{k})^{G} vanishes along Σ to order at least k · e(G).

Then there is an element t ∈ H^{0}(W, ω_{W}) such that q^{∗}r_{∗}t = s.

étale case normalization Abhyankar

2.6 Mildness

A flat morphism X → B is said to be mild if whenever B′ has rational-Gorenstein singularities and B′ → B dominant, then X ×_{B} B′ has rational-Gorenstein singularities.

2.7 Observation

If X → B mild then X_{B}^{n} → B is mild as well.

2.8 Proposition

Assume X → B is a Gorenstein flat morphism, both B and B′ have rational-Gorenstein singularities and B′ → B dominant. Assume X ×_{B}B′ has rational-Gorenstein singularities. Then X has rational-Gorenstein singularities.

2.9 Theorem (Hochster)

a toric singularity is rational.

2.10 Proposition

a family X → B of nodal curves is mild.

2.11 Proposition

Let X → B be a flat generically smooth projective morphism. Then there is an alteration B′ → B and a modification Y → X ×_{B} B′ such that Y → B is mild. ^{Galois}