References & Citations
Mathematics > Algebraic Geometry
Title: Nef and Effective cones of some Quot Schemes
(Submitted on 7 May 2024 (v1), last revised 15 May 2024 (this version, v2))
Abstract: Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g(C)\geqslant 3$ (respectively, $g(C)=2$). Fix integers $r,k$ such that $2\leqslant k\leqslant r-2$, (respectively, $3\leqslant k\leqslant r-2$). Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(\mathcal{O}^{\oplus r}_C, k, d)$ be the Quot scheme parametrizing rank $k$ and degree $d$ quotients of the trivial bundle of rank $r$. Let $\mathcal{Q}_L$ denote the closed subscheme of the Quot scheme parametrizing quotients such that the quotient sheaf has determinant $L$. It is known that $\mathcal{Q}_L$ is an integral, normal, local complete intersection, locally factorial scheme of Picard rank 2, when $d\gg0$. In this article we compute the nef cone, effective cone and canonical divisor of this variety when $d\gg0$. We show this variety is Fano iff $r=2k+1$.
Submission history
From: Ronnie Sebastian [view email][v1] Tue, 7 May 2024 07:15:33 GMT (23kb)
[v2] Wed, 15 May 2024 13:10:16 GMT (16kb)
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