Families of sheaves and automorphisms

Question

Let $X$ be a scheme, $S$ a $K3$ surface and $F$ a flat family of coherent sheaves on $S$ parametrized by $X$. Let us assume that for every $x\in X$ $F_x$ is locally free, has fixed Chern classes and satisfies $h^1(F_x)=h^2(F_x)=0$. Does there exist a scheme $A$ together with a morphism $p:A\to X$ such that $p^{-1}(x)$ is isomorphic to $Aut(F_x)$ for every $x\in X$? (by $F_x$ I mean $F|_{X_x}$ with $X_x$ the fiber of the projection $X\times S\to X$ over the point $x$)

Answer 1

There are standard ways of constructing this kind of objects, but I can't immediately think of a reference, so here it goes:

Let $p:Y\to X$ be a projective map (in your case, $Y=S\times X$), let $F$ and $G$ be coherent sheaves on $Y$ with $G$ flat over $X$. Then there is a scheme $H$ of finite type over $X$ whose fiber over $x$ is the space $Hom(F_x,G_x)$. (More concretely, $H$ represents certain natural functor.) In your case, $F=G$, and your $A$ is an open subscheme of $H$ consisting of invertible homomorphisms, i.e., automorphisms.

Sketch of construction of $H$: Write $F$ as the cokernel of a map $$d:O(-n_1)^{N_1}\to O(-n_2)^{N_2}$$ for $n_1,n_2\gg 0$. Morphisms from $O(-n)$ to $G$ correspond to sections of $p_*(G(n))$; by assumptions, it is a vector bundle if $n\gg0$. Let $G_n$ be the total space of this vector bundle. Then $d$ induces a map $$(G_{n_2})^{N_2}\to (G_{n_1})^{N_1},$$ and $H$ is the preimage of the zero section.

Answer 2

You most certainly need to assume that $S$ is proper (or something similar); even when $X$ is the spectrum of a field but $S$ is affine, say, you will not get what you want (unless you accept something silly like taking $\mathrm{Aut}(F)$ as the disjoint union of copies of $S$). Assuming some properness you are still probably in trouble unless you assume that $p_\ast F$ (where $p\colon X\times S\rightarrow X$ is the projection) is locally free and commutes with scalar extension. That there is some problem can be seen by letting $X=\mathrm{Spec}R$, $R$ a discrete valuation ring, $S=\\mathrm{Spec}\mathbb Z$ and $F$ the sheaf on $X\times S=X$ associated to $R\bigoplus R/m$, where $m$ is the maximal ideal. Hence we are looking for a group scheme over $X$ whose generic fibre is the multiplicative group and whose special fibre is $\mathrm{GL}_2$ and for which the specialisation of the multiplication group is the natural subgroup of $\mathrm{GL}_2$. Unless I am mistaken such a group scheme does exist (you don't require $A$ to be a group scheme and you can certainly construct such a scheme). It doesn't look very nice however and things become much worse if you let $X$ have higher dimension and then if $p$ is not an isomorphism you also have to fight with the fact that $p_*F$ may not commute with base change which makes things even worse.

Hence even though I don't have a specific counter example to the existence of your $A$ fulfilling your very weak conditions, if you do not assume that $S$ is proper and that $p_*F$ is locally free and commute with base change I doubt that you can get an answer that you can actually use.