(page 1) Chapter 11

flat family

Hilbert polynomial

Mumford’s Regularity Theorem Grauert’s Theorem

Universal Families of Curves

Let X be a smooth projective surface over a field k. Let D ⊂ X ×T be a flat family of curves. We have the usual exact sequence

(11.1)  0 → O_{X×T} (−D) → O_{X×T} → O_{D} → 0,

which specializes on each fibre to the corresponding sequence with X × T replaced by X_{t} = X ×_{k} k(t) and D replaced by D_{t}. Now we can assume that all the fibres have the same Hilbert polynomial

P (m) = χ(O_{X_{t}}(−D_{t} + m)).

By Mumford’s Regularity Theorem, there exists a positive integer m_{0}, independent of t, such that O_{X_{t}}(−D_{t}) is m_{0}-regular and such that H^{i}(O_{X_{t}}(m_{0})) = 0 for i > 0. The regularity ensures in particular that the higher cohomology groups H^{i}(X_{t}, O_{X_{t}}(−D_{t} + m_{0})) vanish for i > 0. Thus,

P (m_{0}) = χ(O_{X_{t}}(−D_{t} + m_{0})) = dim(H^{0}(O_{X_{t}}(−D_{t} + m_{0}))).

Therefore, twisting the sequence (11.1) by m_{0} and taking cohomology yields a short exact sequence

0 → H^{0}(O_{X_{t}}(−D_{t} + m_{0})) → H^{0}(O_{X_{t}}(m_{0})) → H^{0}(O_{D_{t}}(m_{0})) → 0.

By extending the cohomology exact sequence further to the right, one also concludes that H^{i}(O_{D_{t}}(m_{0})) = 0 for i > 0. In particular, we see that

dim(H^{0}(O_{D_{t}}(m_{0}))) = χ(O_{D_{t}}(m_{0}))

is also independent of t, since it is the value of a Hilbert polynomial in a flat family.

Let π : X × T → T denote the projection map. Since the dimensions of the cohomology groups along the fibres are independent of t, we can use Grauert’s Theorem to conclude

• R^{i}π_{∗}(O_{X×T} (−D + m_{0})) = 0 for i > 0.

• π^{∗}π_{∗}(O_{X×T} (−D + m_{0})) → O_{X×T} (−D + m_{0}) is surjective.
• π_{∗}(O_{D}(m_{0})) is locally free of rank r = χ(O_{X}(m_{0})) − P (m_{0}).
◯^{c}Coombes 1996  1

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2 Chapter 11: Universal Families of Curves

Künneth Formula Grassmann variety Künneth Formula effective Cartier divisor divisor, relative Grassmann variety

Twisting sequence (11.1) by m_{0} and pushing down along π gives a short exact sequence

0 → π_{∗}(O_{X×T} (−D + m_{0})) → π_{∗}(O_{X×T} (m_{0}))⟶^{σ}π_{∗}(O_{D}(m_{0})) → 0.

By the Künneth Formula, the middle term is

π_{∗}(O_{X×T} (m_{0})) = O_{T} ⊗_{k} H^{0}(O_{X}(m_{0})).

Fix a basis e_{0}, …, e_{N} of H^{0}(X, O_{X}(m_{0})). From our family of curves D on X×T, we have produced a locally free sheaf π_{∗}(O_{D}(m_{0})) of rank r and N + 1 global sections s_{i} = σ(1⊗e_{i}) that span the sheaf. This collection of data is, by definition, a T valued point of the Grassmann variety G_{r,N}. It is clear that our construction is compatible with pullbacks, so we have actually constructed a natural transformation of functors

Φ : Curves_{X}^{P} → G_{r,N}.

Now suppose we are given any T -valued point f : T → G_{r,N} of the Grassmann. Unsurprisingly, f gives us a locally free sheaf E on T and N + 1 sections s_{0}, …, s_{N} that span E. So, we have a surjective homomorphism

O_{T} ⊗_{k} H^{0}(O_{X}(m_{0}))⟶^{σ}E → 0.

Let K denote the kernel of σ. Pulling back along the projection map gives an exact sequence

π^{∗}(K) → π^{∗}(O_{T} ⊗_{k} H^{0}(O_{X}(m_{0}))) → π^{∗}E → 0.

By the Künneth Formula, there is a natural map from the term in the middle to O_{X×T} (m_{0}). Compose this natural map with the first map in the exact sequence. Twisting by −m_{0} yields a homomorphism

ρ : π^{∗}(K)(−m_{0}) → O_{X×T} .

Define I = Im(ρ). This is an ideal sheaf on X × T, so it must define a closed subscheme of X × T. Thus, one has another functor

Ψ : G_{r,N} → Subschemes_{X}.

Proposition 11.2. The composite Ψ∘Φ : Curves_{X}^{P} → Subschemes_{X} is the natural inclusion of curves on X in the space of all subschemes on X.

Proof: We start with a relative effective Cartier divisor D on X ×T. The map into the Grassmann variety associates to D the locally free sheaf E = π_{∗}(O_{D}(m_{0})). The kernel of the map from O_{T} ⊗_{k}H^{0}(O_{X}(m_{0})) onto E is just K = π_{∗}(O_{X×T} (−D+m_{0})). Since this sheaf is spanned by global sections, we can conclude that I is just the original ideal sheaf O_{X×T} (−D).

In abstract terms, we are now in the following situation. We have • a pair of functors A, B : SCH/k → SET ,

• a scheme G, and

◯^{c}Coombes 1996 flat family Hilbert polynomial Flattening Stratification

• a pair of natural transformations

A⟶^{Φ}h_{G}⟶^{Ψ}B

such that the composite is an injection.

Proposition 11.3. Assume A, B, G, Φ and Ψ are as above. Assume in addition that for all schemes T and all elements β ∈ B(T) there exists a subscheme Y ⊂ T with the following property: A morphism f : S → T factors through Y if and only if f^{∗}(β) ∈ A(T) ⊂ B(T). Then there exists a subscheme G_{0} ⊂ G such that A = h_{G_{0}} and Φ is the natural transformation induced by the inclusion of schemes.

Proof: Take β ∈ B(T) to be the point corresponding to the identity map from G to itself. By hypothesis, there exists a subscheme Y ⊂ G such that a morphism f : T → G factors through Y if and only if f^{∗}(β) lies in A(T). Since β corresponds to the identity map, this simply says that a T -point of G lies in A(T) if and only if f : T → Y. The result folows by taking G_{0} = Y.

The hard part, of course, is to verify this abstract property in the concrete context of curves and subschemes on an algebraic surface. The condition translates to the following property: For every closed subscheme Z ⊂ X × T, there exists a subscheme Y ⊂ T such that a morphism f : S → T factors through Y if and only if Z ×_{T} S is a flat family of curves over S whose ideal sheaf has Hilbert polynomial P. We need the following result.

Theorem 11.4. (Flattening Stratification) Let F be a coherent sheaf on P^{n} × T where T is a noetherian scheme. Then there exists a finite covering collection T_{1}, …, T_{m} of disjoint, locally closed subschemes of T such that for all morphisms f : S → T from a noetherian scheme S, the pullback F_{f} = (1 × f)^{∗}F is flat over S if and only if f factors through the disjoint union ∐ T_{i}.

Proof: We are only going to sketch a proof of this result. For further details, consult Mumford @@464@@.

The first step is to do the case n = 0. In this case, F is a coherent sheaf on T itself. The pullback F_{f} = f^{∗}(F) is flat if and only if it is locally free. Since the function e(t) = dim_{k(t)}(F_{t} ⊗ k(t)) is upper semicontinuous, the sets Z_{e} = {t ∈ T : e(t) = e} are locally closed subsets in T. The main point is to show that the Z_{e} have a natural subscheme structure and so can be used as the flattening stratification in this case.

For the gernal case, one must first do some hard commutative algebra to show that if f : X → Y is a morphism of finite type between noetherian schemes with Y integral, and if F is a coherent sheaf on X, then there is an open subscheme of Y over which F is flat. This result implies that there is a finite covering set of locally closed subschemes of T over which the restriction of F is flat. Now consider the sheaves E_{m} = (p_{T})_{∗}(F(m)). It can be shown that

◯^{c}Coombes 1996 Flattening Stratification Cartier divisor divisor, Cartier divisor, Cartier Cartier divisor Nakayama’s Lemma

• There is a uniform m_{0} such that H^{i}(P_{t}^{n}, F_{t}(m)) = 0 for all i > 0, all m ≥ m_{0}, and all t. In this range, then, E_{m} ⊗ k(t) = H^{0}(P_{t}^{n}, F_{t}(m)).

• Only finitely many different polynomials occur as the Hilbert polynomials fo the sheaves F_{t}.

• F_{f} is flat if and only if f^{∗}(E_{m}) is flat for all m ≥ m_{0}. Now a flattening stratification for F is the “greatest common divisor” of the

flattenting stratifications for all the E_{m} with m ≥ m_{0}. These stratifications can be combined into a finite cofvering set by considering the finite list of distinct Hilbert polynomials.

By the theorem on Flattening Stratifications, we can find a subscheme Y ⊂ T such that Z ×_{T} S is flat over S, with Hilbert polynomial

χ(O_{Z×_{T} S}(n)) = χ(O_{X}(n)) − P (n)

if and only if f : S → T factors through Y. We still need to find out when Z ×_{T} S is actually a Cartier divisor.

Lemma 11.5. Let Z ⊂ X × T be a closed subscheme that is flat over T. Let t ∈ T be a closed point such that Z_{t} is a curve on X. Then there exists an open neighborhood U of t in T such that Z ∩ X × U is a Cartier divisor on U.

Proof: Since π : X × T → T is a proper map, it is closed. So, it suffices to prove that there is an open neighborhood V of X × {t} in which Z is a Cartier divisor. Let x ∈ X × T be any point with π(x) = t. Let I_{x} ⊂ O_{x} be the ideal defining Z locally at x, and let M_{t} ⊂ O_{t} be the maximal ideal in the local ring at t. Now the local ring of x on the fibre X × {t} is just O_{x}/M_{t}O_{x}. Since Z_{t} is a Cartier divisor, its ideal on the fibre is locally principal. So, we can find an element f ∈ O_{x} such that

I_{x} + M_{t}O_{x} = (f) + M_{t}O_{x}.

By modifying f if necessary, we can assume that f ∈ I_{x}. Now consider the exact sequence

0 → I_{x}/(f) → O_{x}/(f) → O_{x}/I_{x} → 0.

Since Z is flat over T, tensoring this sequence with k(t) yields

0 → I_{X}/(f) ⊗ k(t) → O_{x}/((f) + M_{t}O_{x}) → O_{x}/(I_{x} + M_{t}O_{x}) → 0.

By the earlier equality, we see that I_{x}/(f) ⊗ k(t) = 0. By Nakayama’s Lemma, we see that I_{x} = (f). Hence, Z is a Cartier divisor at the point x, and hence it must be a Cartier divisor in a neighborhood of x.

Modulo the Theorem on Flattening Stratifications, we have proven the following result.

◯^{c}Coombes 1996 Weil divisor divisor, Weil effective Cartier divisor divisor, relative Hilbert polynomial

Theorem 11.6. If X is a smooth projective surface over a field, then Curves_{X} is represented by a k-scheme that is locally of finite type.

In particular, there is a scheme that can be thought of as the universal family of curves on X. We do not know, however, if the scheme Curves_{X} has any nice properties. Is it projective? quasiprojective? nonsingular? normal?

Proposition 11.7. The scheme Curves_{X}^{P} is a closed subscheme of G_{r,N}, and it is, therefore, a projective scheme.

Proof: To save on ink, let’s write Y = Curves_{X}^{P}. Take Ȳ to be the set-theoretic closure of Y in G_{r,N}. If Y is not closed, then we can take a closed point y ∈ Ȳ − Y. Fix the following items:

• An affine open neighborhood U = Spec(R) of y in Ȳ.

• The maximal ideal M ⊂ R defining y.

• An ideal I ⊂ M defining the closed set (Ȳ − Y) ∩ U. Now there exists a prime ideal P such that P ⊂ M, I ⊄ P, and dim(R_{P} /P R_{P}) = 1. Let S be the integral closure of the domain R/P in its field of fractions, and write C = Spec(S). Then C is a one-dimensional nonsingular variety, and the obvious homomorphism from R to S defines a morphism

f : C → U ⊂ Ȳ .

Let P̄ be a prime ideal of S lying over M · (R/P). Then P̄ defines a closed point c ∈ C with y = f(c). Let C_{0} = f^{−1}(Y), and let f_{0} be the restriction of f to C_{0}. By construction, f_{0} is a C_{0}-valued point of Y which is not the restriction of any C-valued point of Y. (Look at the closure of the graph of f_{0}.)

We will show that this is impossible. After all, Y = Curves_{X}^{P}. Therefore, f_{0} defines a family of curves D_{0} ⊂ X × C_{0} which is not the restriction of a family of curves on C. But both C and X are nonsingular, and Cartier divisors on the nonsingular product X ×C are the same thing as Weil divisors. We can simply take the closure of the Weil divisor D_{0} to get an effective Weil divisor on X × C. This closure cannot contain any complete fibres of the projection onto C, so it must, in fact, be an effective Cartier divisor relative to C. Since C is connected, the Hilbert polynomial must be constant, so the new piece inserted on the extra points of C must have the same Hilbert polynomial as the old pieces.

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