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arXiv 2007.02535math.AG
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节曲线上向量丛模空间的投射Poincaré丛和Picard丛

Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

C. Arusha, Usha N. Bhosle, Sanjay Kumar Singh

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29 pages, comments are welcome!
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摘要：

设$U^{'s}_L(n,d)$是秩为$n$且行列式为$L$的稳定向量丛的模空间，其中$L$是节曲线$Y$上的度为$d$的固定线丛。证明了在适当的极化下，$Y \times U^{'s}_L(n,d)$上的投影Poincare丛和$U^{'s}_L(n,d)$上的投影Picard丛是稳定的.对于一个非奇异点$x \in Y$，我们表明，限制的投影庞加莱丛$x \times U^{'s}_L(n,d)$是稳定的任何极化。证明了对于算术亏格$g\ge 3$和$g=n=2, d$奇，秩为$n$且行列式$L$为$d$的半稳定向量丛的模空间$U'_L(n,d)$的Picard群同构于$\mathbb{Z}$ .

英文摘要：

Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L(n,d)$ of semistable vector bundles of rank $n$ with determinant $L$ of degree $d$ is isomorphic to $\mathbb{Z}$.

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