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.



Normalized rank 2 vector bundle over a curve
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?


