Let $S$ be a $K3$ surface and $X$ the moduli space of some stable sheaves on it. Let $G$ be the universal family on $X\times S$ and $F$ the ideal of section of $X\times S\to X$. Knowing that for every $x\in X$ Ext$^1(F_x,G_x)$ has dimension 1 and Hom$(F_x,G_x)=0$, can I find an $\mathcal{O}_{X\times S}$-module $E$ flat over $X$ such that $E_x\in$Ext$^1(F_x,G_x)$ for every $x\in X$?
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1$\begingroup$ Are you asking for a bound? Because I don't see that there is one, given this information. Let $S$ be the spectrum of a field for simplicity. There is an exact sequence $$0\to H^1(\mathcal{E}^0)\to Ext^1(F,G)\to H^0(\mathcal{E}^1)$$ where $\mathcal{E}^i=\mathcal{E}xt^i(F,G)$. You know that $\mathcal{E}^1$ is a line bundle, but this is not enough to control the dimensions above. $\endgroup$– Donu ArapuraOct 12, 2010 at 11:51
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$\begingroup$ I have changed the question in order to be more specific. $\endgroup$– ginevra86Oct 12, 2010 at 12:29
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$\begingroup$ OK, that's more precise. But you don't really mean that $E_x\in Ext^1$ do you? $\endgroup$– Donu ArapuraOct 12, 2010 at 12:43
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$\begingroup$ I mean that $E_x$ is an extension of $G_x$ by $F_x$. Moreover, I can add the hypothesis that the group Hom$(F_x,G_x)=0$ for every $x\in X$. $\endgroup$– ginevra86Oct 12, 2010 at 12:51
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