Families of sheaves and automorphisms 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 $x$)
 A: 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.
A: 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.
