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Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g$. All the semistable rank 2 vector bundle $E$ with a fixed determinant bundle $L$ form a projective variety $\mathcal{M}_{2, L}$of dimension 3(g-1).

Question: What is the dimension of the locus of normalized ones in $\mathcal{M}_{2, L}$? Can we calculate its class in Chow ring? By "E is normalized", I mean in Hartshorne that $h^0(E)>0$, but $h^0(E\otimes M)=0$ for any negative degree line bundle $M$.

Thanks for reading.

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If $h^0(E) > 0$ then there is a nonzero global section $s$. Note that $s$ can not vanish in a point. Indeed, if $s$ vanishes at point $p \in C$ then it also gives a section of $E(-p) = E\otimes O_C(-p)$ and $O_C(-p)$ is a line bundle of negative degree.

The section $s$ can be considered as a morphism $O_C \to E$, and as we already checked its rank is everywhere 1, hence the cokernel is locally free of rank 1. Thus we have an exact sequece $$ 0 \to O_C \to E \to L' \to 0, $$ where $L'$ is a line bundle. Comparing determinants we see that $L' = L$. So, $E$ is an extension of $L$ by $O_C$. These are parameterized by $P(Ext^1(L,O_C)) = P(H^1(L^*))$ and you can easily compute the dimension. Of course you have to add the condition that $H^0(E\otimes M) = 0$ for all $M$ of negative degree. This condition will depend on the properties of $L$. What do you know about it?

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  • $\begingroup$ @Sasha, say $\deg L>0$. Then $\dim\mathbb{P}(H^1(L^*))=\deg L+g-2$ should be less than or equal to $\dim\mathscr{M}_{2, L}=3g-3$, i.e. $\deg L\le 2g-1$. what else can we say to approach the problem? $\endgroup$ – Thunder Nov 5 '12 at 0:35

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