Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf

Question

I have been trying to learn some deformation theory, and came across the following in a paper:

The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with $H^0(X,f^*(\mathcal{T}_Y))$.

I would like to understand a proof of this. I understand some simple facts, like $H^1(X,\mathcal{T}_X)$ being bijective with the first order deformations of $X$. The reference given was to a paper of Ravi Vakil's, but I am unfamiliar with anything but the very basics of stacks. Does anyone know of a reference for this fact that doesn't use stacks?

Answer 1

There is actually the following general result.

Let us consider a morphism of algebraic schemes $f \colon X \to Y$, where $X$ is reduced and projective and $Y$ smooth. Then the first order deformations of $f$ leaving the domain and the target fixed are parametrized by $H^0(X, f^* T_Y)$.

For a proof (that does not use stacks!) see Sernesi' book "Deformations of algebraic schemes", in particular Proposition 3.4.2 page 158.

Answer 2

To deform a map $f:X \to Y$ is equivalent to deforming its graph $\Gamma \subset X\times Y$. But a deformation of a subscheme $Z \subset V$ are governed by the normal bundle $N_{Z/V}$ (the tangent space to deformations is $H^0(Z,N_{Z/V})$ and the obstructions are given by $H^1$), this can be found in Grothendieck's Fondements de la Géometrie Algébrique. So, it remains to note that $N_{\Gamma/X\times Y} = f^*T_Y$.

Answer 3

For a more elementary reference, try Debarre's "Higher dimensional algebraic geometry." This book is a great place to get started learning some algebraic geometry beyond Hartshorne, and basic deformation theory plays a key role.

The fact you are asking about is Proposition 2.4; Chapter 2 gives a good overview of similar types of results, and much of the rest of the book investigates how these types of results can be applied.

Answer 4

Kollar's 'Rational Curves on Algebraic Varieties' is a good reference.

Answer 5

Eiji HORIKAWA, On deformations of holomorphic maps I, J. Math. Soc. Japan Volume 25, Number 3 (1973), 372-396.