Extension of a first order deformation of a sheaf

Question

Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.

Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.

Assume all obstructions classes belonging to $F$ vanish, e.g. $Ext^2(F,F)=0$ and we have a nontrivial element $x \in Ext^1(F,F)$, i.e. a deformation $G$ of $F$ over $k[\epsilon]$.

Can we extend this deformation to a curve? More precisely:

Can we find a deformation $\mathcal{F}$ of $F$ over a smooth connected curve $C$, such that $\mathcal{F}$ gives back $G$ under the pullback induced by $Spec(k[\epsilon])\rightarrow C$?

If yes, is this merely an existence result or can we construct the sheaf $\mathcal{F}$ and the curve $C$?

Answer 1

Let us assume that we are working over $\mathbb{C}$.

If $F$ is a stable sheaf then the answer to your first question is yes.

In fact, in this case there exists a quasi-projective moduli space $M$ and, since $Ext^2(F, F)=0$, at the point $[F]$ this moduli space is smooth, of dimension $Ext^1(F, F)$. Moreover, the tangent space $T_{[F]}M$ is canonically identified with $Ext^1(F, F)$.

By the (uni)versal property of the moduli space, the first-order deformation corresponding to $x \in Ext^1(F, F)$ is obtained by pull-back of the universal family $\mathcal{V} \to M$ under a uniquely determined morphism $v \colon \textrm{Spec}(k[\varepsilon]) \to M$, such that $v(\textrm{closed point})=[F]$.

Since $M$ is smooth at $[F]$, we can choose a smooth curve $C$ with a marked point $p$ and an embedding $\iota \colon C \to M$ such that

1. $\iota(p)=[F]$;

2. the tangent vector of $\iota(C)$ at the point $[F]$ equals $x$.

Therefore pulling back the universal family $\mathcal{V} \to M$ via $\iota \colon C \to M$ we obtain a deformation $\mathcal{F}$ of $F$ over the curve $C$ with the desired properties.

Of course this is only an existence result, and I think there are no constructive methods available in general.