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Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $-1$: $$ G(\mathcal{F},(1,-1)):=\{\mathcal{L}\subset \mathcal{F}:\mathcal{L}\text{ is a line bundle of degree $-1$}\}. $$ It naturally embeds in the degree $-1$ Picard variety of $X$, $\mathrm{Pic}^{-1}(X)$, sending $\mathcal{L}$ to the corresponding point of $\mathrm{Pic}^{-1}(X)$. If $g=2$ and $\mathcal{F}$ is stable, this is a one-dimensional subvariety of $\mathrm{Pic}^{-1}(X)$ (This is for example mentioned in the first paragraph of Lange, Narasimhan, Maximal subbundles of rank two vector bundles on curves).

Is it known what this subvariety is? It looks like there is a lot of litterature concerning this and theta divisors but I have not been able to find information on the topology of these subvarieties, more precisely on their cohomology.

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  • $\begingroup$ There is something wrong in your formulation. Since $\mathcal{F}$ is stable, it doesn't contain sub-line bundles of degree $d-1$ when $d\geq2$; and for $d< 2$, the dimension of the subvariety of such bundles in $\operatorname{Pic}^{d-1}(X) $ will depend very much on $d$. You probably mean something different. $\endgroup$
    – abx
    Dec 13, 2021 at 13:26
  • $\begingroup$ Yes, thanks. I had in mind the case $d=0$ for which I hope the question makes sense. I edited the question. $\endgroup$
    – hennlu
    Dec 13, 2021 at 15:08

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For $g=2$,$d=0,$ this is studied in M.S. NARASIMHAN, S. RAMANAN: Moduli of vector bundles on a compact Riemann surface. Ann. of Math. 89, 19-51. It is shown that the divisor is linear equivalent to $2\theta,$ where $\theta$ is the $\theta$-divisor. The latter space is a $\mathbb CP^3$ (as the corresponding space of holomorphic sections is 4-dimensional), and the natural map $\mathcal F\to\mathcal G\in\mathbb CP^3$ gives an isomorphism of the moduli space of (S-equivalence classes) of semi-stable holomorphic rank 2 bundles with $\mathbb CP^3$. The locus of (S-equivalence classes of) strictly semi-stable bundles is the Kummer variety associated to the Jacobian of $X$.

This construction has been generalized by Beauville to degree $1-g$ line subbundles on a surfaces of genus $g$, see for example his overview article: 'Vector bundles on curves and generalized theta functions: recent results and open problems'. I am not aware of any detailed investigation of your $\mathcal G$, for $g>2$.

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  • $\begingroup$ Thanks for this answer! Is it known what the support of a $2\theta$-divisor look like? $\endgroup$
    – hennlu
    Dec 13, 2021 at 15:10
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    $\begingroup$ In the case of a strictly semi-stable bundle $L\oplus L^*$, the divisor is $L\theta+L^*\theta.$ In general, I do not know the answer, but I am sure that certain properties are know. Maybe, a good start is the aforementioned paper by Narasimhan and Ramanan, and the book about compact Riemann surfaces by Narasimhan. $\endgroup$
    – Sebastian
    Dec 13, 2021 at 18:24

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