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Yusuf Mustopa
Yusuf Mustopa
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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 $SU(r,L)$).

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 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 $SU(r,L)$).

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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Yusuf Mustopa
Yusuf Mustopa
  • 1.3k
  • 8
  • 8

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 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 $SU(r,L)$).