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Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is an intersection of 2 4-dimensional quadrics, and it is Fano. If I recall correctly, all moduli spaces of bundles with odd degree on an algebraic curve are fine. My question is: are all fine moduli varieties of VB on an algebraic curve Fano? If not, please give counterexamples. |
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I assume you are asking about $SU(r,L),$ (semistable rank-$r$ bundles with determinant $L$) rather than $U(r,d)$ (semistable rank-$r$ bundles with determinant of degree d). Drezet-Narasimhan showed that even when $SU(r,L)$ is not a fine moduli space, it is locally factorial with Gorenstein singularities, and that its dualizing sheaf is isomorphic to $\mathscr{L}^{-2(r,c_{1}(L))}$ where $\mathscr{L}$ is the (ample) determinant bundle; consequently $SU(r,L)$ is Fano. $U(r,d)$ is then a Fano fibration over the degree-$d$ Picard variety of the underlying curve via the determinant map (the fiber over a degree-$d$ line bundle $L$ is just $SU(r,L)$). |
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