6.1. The spaceMor(Y, X). Grothendieck vastly generalized the preceding con-struction: ifX andY are varieties over a fieldk, withX quasi-projective andY projective, he shows ([G2], 4.c) thatk-morphisms fromY toXare parametrized by a scheme Mor(Y, X) locally of finite type. As we saw in the case Y = P1 and X = PN, this scheme will in general have countably many components.
One way to remedy that is to fix an ample divisorH onX and a polynomial P with rational coefficients: the subscheme MorP(Y, X) of Mor(Y, X) which parametrizes morphismsf :Y →X with fixedHilbert polynomial
P(m) =χ(Y, mf∗H)
is now quasi-projective overk, and Mor(Y, X) is the disjoint (countable) union of the MorP(Y, X), for all polynomials P. Note that whenY is a curve, fixing the Hilbert polynomial amounts to fixing the degree of the 1-cyclef∗Y for the embedding ofX defined by some multiple ofH.
The fact that Y is projective is essential in this construction: the space Mor(A1k,ANk) isnota disjoint union of quasi-projective schemes.
Let us make more precise this notion of parameter space. We ask as above that there be auniversal morphism(also calledevaluation map)
funiv:Y ×Mor(Y, X)→X such that for anyk-schemeT, the correspondance between
• morphismsϕ:T →Mor(Y, X) and
• morphismsf :Y ×T →X obtained by sendingϕto
f(y, t) =funiv(y, ϕ(t))
is one-to-one.
In particular, ifL⊃kis a field extension,L-points of Mor(Y, X) correspond toL-morphismsYL→XL (whereXL=X×SpeckSpecLand similarly forYL).
Examples 6.2 1) The scheme Mor(Speck, X) is just X, the universal mor-phism being the second projection
funiv: Speck×X −→ X.
2) When Y = Speck[ε]/(ε2), a morphism Y → X corresponds to the data of ak-pointx ofX and an element of the Zariski tangent spaceTX,x = (mX,x/m2X,x)∗.
6.3. The tangent space toMor(Y, X). We will use the universal property to determine the Zariski tangent space to Mor(Y, X) at ak-point [f]. This vector space parametrizes by definition morphisms from Speck[ε]/(ε2) to Mor(Y, X) with image [f] ([H1], Ex. II.2.8), hence extensions off to morphisms
fε:Y ×Speck[ε]/(ε2)→X
which should be thought of as first-order infinitesimal deformations off. Proposition 6.4 LetX andY be varieties over a fieldk, withXquasi-projective andY projective, letf :Y →X be ak-morphism, and let[f]be the correspond-ingk-point of Mor(Y, X). One has
TMor(Y,X),[f] 'H0(Y,Hom(f∗ΩX,OY)).
Proof. Assume first that Y and X are affine and write Y = Spec(B) and X= Spec(A) (whereAandBare finitely generatedk-algebras). Letf]:A→B be the morphism corresponding tof, makingBinto anA-algebra; we are looking fork-algebra homomorphismsfε]:A→B[ε] of the type
∀a∈A fε](a) =f(a) +εg(a).
The equalityfε](aa0) =fε](a)fε](a0) is equivalent to
∀a, a0 ∈A g(aa0) =f](a)g(a0) +f](a0)g(a).
In other words, g : A →B must be a k-derivation of the A-moduleB, hence must factor as g : A → ΩA → B ([H1], §II.8). Such extensions are therefore parametrized by HomA(ΩA, B) = HomB(ΩA⊗AB, B).
In general, cover X by affine open subsets Ui = Spec(Ai) and Y by affine open subsets Vi = Spec(Bi) such that f(Vi) is contained in Ui. First-order extensions off|Vi:Vi →Ui are parametrized by
gi ∈HomBi(ΩAi⊗AiBi, Bi) =H0(Vi,Hom(f∗ΩX,OY)).
To glue these, we need the compatibility condition gi|Vi∩Vj =gj|Vi∩Vj,
which is exactly saying that thegi define a global section onY. In particular, whenX is smooth along the image off,
TMor(Y,X),[f]'H0(Y, f∗TX).
Example 6.5 When Y is smooth, the proposition proves that H0(Y, TY) is the tangent space at the identity to the group of automorphisms of Y. The image of the canonical morphismH0(Y, TY)→H0(Y, f∗TX) corresponds to the deformations off by reparametrizations.
6.6. The local structure of Mor(Y, X). We prove the result mentioned in the introduction of this chapter. Its main use will be to provide a lower bound for the dimension of Mor(Y, X) at a point [f], thereby allowing us in certain situations to produce many deformations off. This lower bound is very accessible, via the Riemann-Roch theorem, whenY is a curve (see 6.11).
Theorem 6.7 Let X and Y be projective varieties over a field k and let f : Y →X be a k-morphism such that X is smooth along f(Y). Locally around [f], the schemeMor(Y, X)can be defined byh1(Y, f∗TX)equations in a smooth scheme of dimension h0(Y, f∗TX). In particular, any (geometric) irreducible component ofMor(Y, X)through [f] has dimension at least
h0(Y, f∗TX)−h1(Y, f∗TX).
In particular, under the hypotheses of the theorem, a sufficient condition for Mor(Y, X) to be smooth at [f] is H1(Y, f∗TX) = 0. We will give in 6.12 an example that shows that this condition is not necessary.
Proof. Locally around thek-point [f], thek-scheme Mor(Y, X) can be defined by certain polynomial equations P1, . . . , Pm in an affine spaceAnk. The rank rof the corresponding Jacobian matrix ((∂Pi/∂xj)([f])) is the codimension of the Zariski tangent spaceTMor(Y,X),[f] in kn. The subvariety V ofAnk defined byr equations among the Pi for which the corresponding rows have rankr is smooth at [f] with the same Zariski tangent space as Mor(Y, X).
Letting hi = hi(Y, f∗TX), we are going to show that Mor(Y, X) can be locally around [f] defined by h1 equations inside the smooth h0-dimensional variety V. For that, it is enough to show that in the regular local k-algebra R =OV,[f], the ideal I of functions vanishing on Mor(Y, X) can be generated byh1 elements. Note that since the Zariski tangent spaces are the same, I is contained in the square of the maximal idealm ofR. Finally, by Nakayama’s
lemma ([M], Theorem 2.3), it is enough to show that thek-vector space I/mI has dimension at mosth1.
The canonical morphism Spec(R/I) → Mor(Y, X) corresponds to an ex-tensionfR/I : Y ×Spec(R/I) → X of f. Since I2 ⊂ mI, the obstruction to extending it to a morphismfR/mI :Y ×Spec(R/mI)→X lies by Lemma 6.8 below (applied to the idealI/mI in thek-algebra R/mI) in
H1(Y, f∗TX)⊗k(I/mI).
Write this obstruction as
h1
X
i=1
ai⊗¯bi,
where (a1, . . . , ah1) is a basis for H1(Y, f∗TX) and b1, . . . , bh1 are in I. The obstruction vanishes modulo the ideal (b1, . . . , bh1), which means that the mor-phism Spec(R/I)→Mor(Y, X) lifts to a morphism Spec(R/I0)→Mor(Y, X), whereI0=mI+(b1, . . . , bh1). The image of this lift lies in Spec(R)∩Mor(Y, X), which is Spec(R/I). This means that the identityR/I→R/I factors as
R/I→R/I0−→π R/I,
whereπis the canonical projection. By Lemma 6.9 below (applied to the ideal I/I0 in thek-algebraR/I0), sinceI⊂m2, we obtain
I=I0 =mI+ (b1, . . . , bh1),
which means thatI/mI is generated by the classes ofb1, . . . , bh1. We now prove the two lemmas used in the proof above.
Lemma 6.8 Let R be a noetherian local k-algebra with maximal ideal m and residue field k and let I be an ideal contained in m such that mI = 0. Let f :Y →X be ak-morphism and letfR/I :Y×Spec(R/I)→X be an extension off. Assume X is smooth along the image off. The obstruction to extending fR/I to a morphism fR:Y ×Spec(R)→X lies in
H1(Y, f∗TX)⊗kI.
Proof. In the case where Y andX are affine, and with the notation of the proof of Proposition 6.4, we are looking fork-algebra liftingsfR] fitting into the diagram
B⊗kR
A
fR] ::
fR/I]
// B⊗kR/I.
BecauseX = Spec(A) is smooth along the image off andI2= 0, such a lifting exists,1and two liftings differ by ak-derivation ofAintoB⊗kI,2 that is by an element of
HomA(ΩA, B⊗kI) ' HomA(ΩA, B⊗kI) ' HomB(B⊗kΩA, B⊗kI) ' H0(Y,Hom(f∗ΩX,OY))⊗kI ' H0(Y, f∗TX)⊗kI.
To pass to the global case, one needs to patch up various local extensions to get a global one. There is an obstruction to doing that: on each intersectionVi∩Vj, two extensions differ by an element of H0(Vi∩Vj, f∗TX)⊗kI; these elements define a 1-cocycle, hence an element in H1(Y, f∗TX)⊗kI whose vanishing is necessary and sufficient for a global extension to exist.3 Lemma 6.9 Let A be a noetherian local ring with maximal ideal m and let J be an ideal in Acontained in m2. If the canonical projectionπ:A→A/J has a section,J = 0.
Proof. Letσbe a section ofπ: ifaandbare inA, we can writeσ◦π(a) =a+a0 andσ◦π(b) =b+b0, where a0 andb0 are in I. Ifaandbare in m, we have
(σ◦π)(ab) = (σ◦π)(a) (σ◦π)(b) = (a+a0)(b+b0)∈ab+mJ.
SinceJ is contained in m2, we get, for anyxin J, 0 =σ◦π(x)∈x+mJ,
henceJ ⊂mJ. Nakayama’s lemma ([M], Theorem 2.2) impliesJ = 0.