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?



Universal family over the moduli space of simple sheaves on a K3 surface
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


$v(E)=ch(E).\sqrt{Td(X)}=(rk(E),c\_1(E),rk(E)+c\_1^2(E)/2-c\_2(E))$.


If the Mukai-vector $v \in H^\*(X,\mathbb{Z})$ is indivisible (i.e. there is a $v'$ with $< v.v' >=1$ for the Mukai-pairing) 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.


