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?
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$\begingroup$ So what you are actually asking is: under what condition is $M$ a fine moduli space? Or? $\endgroup$– Daniel LarssonOct 5, 2010 at 12:57
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1 Answer
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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.
answered Oct 5, 2010 at 13:20
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$\begingroup$ 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 $g.c.d.(rk,c_1.H,c_1^2/2-c_2)=1$. 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
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$\begingroup$ 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