Given $S$ a $K3$ surface and $M$ the moduli space of simple sheaves of rank 2 and fixed Chern classes on $S$, under which conditions does a universal family on $S\times M$ exist?
1 Answer
Let us introduce the Mukai pairing on the cohomology $H^*(X,\mathbb{Z})=H^0\oplus H^2\oplus H^4$ of a K3 surface X, as
$<(r,l,s).(r',l',s')>=l.l'rs'r's$
where $l.l'$ is the intersection pairing on $H^2$. The Mukai vector of a sheaf $E$ is defined to be
$v(E)=ch(E).\sqrt{Td(X)}=(rk(E),c_1(E),rk(E)+c_1^2(E)/2c_2(E))$.
If the Mukaivector $v \in H^*(X,\mathbb{Z})$ is indivisible (i.e. there is a $v'$ with $< v.v' >=1$ for the Mukaipairing) then there is a polarization $h$ such that the Moduli space $M_h(v)$ is fine.
This criterion is due to Mukai: "On the Moduli space of Bundles on K3 surfaces, I" (Tata Inst. Fund. Res. Stud. Math., 11, Tata Inst. Fund. Res., Bombay, 1987). You can find it also in the famous book by Huybrechts and Lehn.

$\begingroup$ In the book you mentioned I found that there always exists a quasiuniversal family while a sufficient condition for the existence of a universal family over $M_H(v)$ is that $g.c.d.(rk,c_1.H,c_1^2/2c_2)=1$. Do you agree with this condition? Does it also implies the existence of a universal family over the moduli space of SIMPLE sheaves? $\endgroup$ Oct 9, 2010 at 15:36

$\begingroup$ Yes, this is equivalent. And no, for simple sheaves, U am not sure, this all concernes stable sheaves only. Sorry I missed this point in your question. $\endgroup$ Oct 29, 2010 at 1:32