MODULI SPACES AND DEFORMATION THEORY, CLASS 11

RAVI VAKIL

Contents

1. First order deformations of nonsingular varieties X/k 1 2. Artin rings 2

1. First order deformations of nonsingular varieties X/k Definition. First-order deformations of f : X → k are precisely fibered dia-

grams of the form

X → X̃
↓  f ↓
Spec k → Spec k[ϵ]/ϵ^{2}

where f is flat.

(Note that the left side is X/k; automorphisms of X don’t come into it!) Denote these Def(X/k). I’ll define deformations (with no adjective) later today. Exercise. Suppose we have some nice moduli stack, e.g. M_{g}. Sow that there is

a bijection between first-order deformations and the tangent space to the Deligne-Mumford stack.

Note that the tangent space to a Deligne-Mumford stack has a natural k-vector space structure, but it isn’t clear that these diagrams do!

Theorem. Def(X/k) is naturally in bijection with H^{1}(X, T_{X}). (Note that the right side also has a vector space structure!) Proof in a few minutes. Exercise. If X is a nonsingular curve of genus at least 2, then h^{1}(X, T_{X}) = 3g−3. We’ll later see that this means that M_{g} is smooth.

Date: Tuesday, October 17, 2000.

Remark. Note that the automorphism group of the curve acts on H^{1}(X, T_{X}). (Make geometric comment about M_{g}.)

Flatness lemma. (Eisenbud Cor. 6.2, p. 163 — this is one of the five basic things one should know about flatness.) If M is a k[ϵ]/ϵ^{2}-module, then M is flat iff

M/ϵM →^{ϵ×} ϵM

is an isomorphism. (Clearly it is surjective. This states that multiplication doesn’t kill any more than necessary.)

Using the above, we see that first-order deformations of X are precisely given by infinitesimals extensions of X by O_{X}, completing the proof of the theorem.

2. Artin rings

Motivation. Show that M_{g} is nonsingular. By way of: M_{g} is nonsingular at a point [C] ∈ M_{g}. Infinitesimal lifting property (and finite type) gives it to us.

More generally, \overline{M}_{g} is nonsingular, and boundary divisors intersect transversely. We’ll show this by understanding the “deformation space of a node”. (Sketch.)

Definition. An Artin ring is a ring satisfying the ascending chain condition. For rings over a field k, this precisely those rings that are finite-dimensional vector spaces. (Draw picture.) A local Artin ring is an Artin ring with only one maximal ideal, e.g. k[x, y, z]/(x^{2}, y^{3}, z^{4} − x − y).

Example: The nth-order formal neighborhood of a k-valued point of a scheme X. Locally, it looks like (A, 𝔪); the neighborhood is A/𝔪^{n+1}.

Let C be the category of local Artin rings over k, with residue field k. In other words, the objects are (A, 𝔪) with residue field k, and morphisms induce isomorphim of the residue field.

Non-example. We’ve lost some Artin rings. For example, the second-order formal neighborhood of (p) in Spec ℤ is (Spec of ) ℤ/p^{2}, which is not a ℤ/p-algebra.

Universal example. These are precisely the nth order formal neighbourhoods of schemes over k that are locally of finite type (or even locally of finite presentation).

Let Ĉ be the category of complete Noetherian local k-algebras, with residue field k, for which A/𝔪^{n} is in C for all n. Notice that C is a full subcategory of C.

Example. A formal neighborhood of a k-valued point of a scheme over k, i.e. the inverse limit of its nth order rings. Usually denoted Spf rather than Spec, to remind you of the limit, and the topology involved.

Denote t_{A}^{∗} by 𝔪/𝔪^{2}; the Zariski cotangent space of Spec A.

Here are some basic facts about Artin rings. I forgot to mention (but will next day): Algebra exercise. A morphism B → A

in C is surjective if and only if the induced map t_{B}^{∗} → t_{A}^{∗} is surjective. C replaced by Ĉ.

We can also check when a morphism C is (formally) smooth. (I’ve put formally in brackets, as quasicompactness is automatic.)

Definition. Suppose G → F, in Ĉ. Then we need to check if

Spf A → Spf F
↓ ↗^{?} ↓
Spf B → Spf G

where B is an extension of A by a square-zero ideal, and B and A are in C. Then we say Spec F → Spec G is smooth.

Don’t be scared by Spf; just do this on rings. I’ve written Spf so as to keep the arrows going in the geometric direction.

Remove square-zero! Replace by surjection B → A, again, to check this, just need to check on tangent spaces.

Similarly, you can define etale (exists exactly one) and unramified (at most one). Thus Spf F → Spf G in Ĉ is smooth (etale, unr) if for all Spf A → Spf B in C,

where B → A is surjective,

Hom(F, B) → Hom(F, A) ×_{Hom_{G,A}} Hom(G, B)

as sets, is surjective, (bijective, injective).

Exercise. It is equivalent to require that B and A are in Ĉ. (Sketch why.) Exercise. (a) Spf F → Spf G is smooth iff F is a power series ring over G. (Etale:

isomorphism; unramified: closed immersion.)

(b) A composition of smooth morphisms is smooth. (Etale, unr.)

(c) If u : Spf F → Spf G and v : Spf G → Spf H and u is surjective and vu is smooth, then v is smooth.

(d) Smoothness is preserved by base change. (Etale, unr.) We talked at length about Spf.