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Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $F$ be a $\mu_L$ semistable rank 2 vector bundle on $X$ (semistability in the sense of Mumford-Takemoto).

We have the projections $p_i:X\times X\longrightarrow X$ for $i=1,2$. Then $L'=p_1^*L\otimes p_2^*L$ is an ample line bundle on $X\times X$.

Are (a) $p_1^*F\oplus p_2^*F$ and (b) $p_1^*F\otimes p_2^*F$ are $\mu_{L'}$-semistable on $X\times X$.

I was first wondering if the pull back $p_i^*F$ is $\mu_{L'}$- semistable. Maruyama's result holds only for finite morphism (pullback of a $\mu_L$-semistable vector bundle is $\mu_{g^*L}$-semistable if $g$ is a finite morphism). Is there some result I can use?

Thanks in advance!

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  • $\begingroup$ Perhaps you can use Mehta-Ramanathan to reduce to Maruyama / Kempf. $\endgroup$ – Jason Starr Nov 24 '15 at 18:05
  • $\begingroup$ Prof. @JasonStarr, sorry I don't really understand. Can you give me more leads? $\endgroup$ – gradstudent Nov 24 '15 at 18:31
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    $\begingroup$ If either of those sheaves is unstable, then there is a destabilizing subsheaf. Now restrict to a sufficiently general complete intersection subvariety $Y$ of $X\times X$ whose projection to each factor $X$ is a finite morphism. The restriction is still destabilized. By your two sheaves are of formation compatible with restriction. By Maruyama / Kempf, the pullbacks of semistable sheaves by finite morphisms are still semistable (in characteristic 0, not in positive characteristic). $\endgroup$ – Jason Starr Nov 24 '15 at 18:43
  • $\begingroup$ Prof. @JasonStarr, can we always find a general complete intersection subvariety $Y$ of $X\times X$ whose projection to each factor $X$ is a finite morphism? $\endgroup$ – gradstudent Nov 30 '15 at 7:47
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    $\begingroup$ Yes, the proof is very similar to the proof of Bertini's theorem. I believe that Gabber - Liu - Lorenzini prove a version of this in a very general setting. However, the case you need is an easy parameter count: with respect to a projective embedding $X\times X \hookrightarrow \mathbb{P}^n$, just consider complete intersections in the linear system $\mathcal{O}(d)$ such that the Hilbert function of the fibers of the projection evaluated at $d$ are strictly larger than the dimension of $X$. $\endgroup$ – Jason Starr Nov 30 '15 at 8:13

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