arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2602.19152math.AG
收藏

皮卡束与扭曲皮卡束在曲线雅可比的限制

Picard bundles and Twisted Picard bundles on the Jacobian of a curve

Indian Statistical Institute, Bangalore（印度统计学院班加罗尔）；Mathematics and Statistics Unit, Indian Statistical Institute, Bangalore（印度统计学院班加罗尔数学与统计单元）

Usha N. Bhosle

备注：
16 pages Lemma 2.5 corrected Remark 5.5 from older version deleted The changes do not affect other results of the paper
发表信息：
Proc. Indian Acad. Sci. (Math. Sci.) (2026) 136:35
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

研究了曲线雅可比上皮卡束的限制及其稳定性，揭示了稳定ACM束的构造与theta divisor的关系。

摘要：

设Y表示一个不可约的射影曲线，其奇点仅包含节点，并且在代数闭域上定义，特征为零。我们研究扭曲皮卡束在Y的紧化雅可比$\overline{J}(Y)$上对嵌入曲线的限制。作为应用，我们显示对于g=2和每个整数r≥3，存在一个二维的稳定ACM束家族在紧化雅可比上，其极限包含皮卡束。我们定义一个嵌入$α_Y$，将雅可比$J(Y)$嵌入到稳定向量束模空间$U^s_Y(n,d)$中，使用扭曲限制$E_Y$，即皮卡束在$Y\subset J(Y)$上的限制。我们证明（在适当条件下）universal bundle$\mathcal{U}$在$Y\times J(Y)$上的限制对于适当极化是稳定的。对于由$E_Y\otimes B$，B是一个次数为b的线束所给定的光滑曲线Y的嵌入，我们显示皮卡束在$U^s_Y(n, d+nb)$上对$J(Y)$的限制对于$b≥2g-1$是$θ$-半稳定，对于$b≥2g$是$θ$-稳定。我们还确定了theta divisor在$U^s_Y(n,d+nb)$上对$J(Y)$的限制与雅可比$\overline{J}(Y)$上的theta divisor$θ$之间的关系。

英文摘要：

Let $Y$ denote an irreducible projective curve with at most nodes as singularities and defined over an algebraically closed field of characteristic zero. We study the restriction of the twisted Picard bundles on the compactified Jacobian $\overline{J}(Y)$ of $Y$ to the embedded curve in $\overline{J}(Y)$. As an application, we show that for $g =2$ and each integer $r \ge 3$, there is a two-dimensional family of stable ACM bundles on the compactified Jacobian which has the Picard bundle in its limit. We define an embedding $α_Y$ of the (generalised) Jacobian $J(Y)$ in the moduli space $U^s_Y(n,d)$ of stable vector bundles on $Y$ using a twisted restriction $E_Y$ of a Picard bundle to $Y \subset J(Y)$. We show that (under suitable conditions) the restriction of the universal bundle $\mathcal{U}$ to $Y \times J(Y)$ is stable for suitable polarisation. For the embedding of a smooth curve $Y$ given by $E_Y \otimes B, B$ a line bundle of degree $b$, we show that the restriction of the Picard bundle on $U^s_Y(n, d+nb)$ to $J(Y)$ is $θ$-semistable for $b \ge 2g-1$ and $θ$-stable for $b \ge 2g$. We also determine the relation between the restriction of the theta divisor on $U^s_Y(n,d+nb)$ to $J(Y)$ and the theta divisor $θ$ on $\overline{J}(Y)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑