Simpson Jacobians of generalized tree-like curves

Simpson Jacobians of reducible curves

For any projective curve $X$ let $\bar{M}^d(X)$ be the Simpson moduli space of pure dimension one rank 1 degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the connected component of $\bar{M}^d(X)$ that contains semistable line bundles can be considered as the compactified Jacobian of $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration,…

The Local Structure of Compactified Jacobians

This paper studies the local geometry of compactified Jacobians constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. The main result is a presentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve. The authors have investigated the geometric and combinatorial properties of these rings in previous work, and consequences for compactified Jacobians are presented in this paper. Similar results are given for the local structure of the universal compactified Jacobian over the…

关于节点曲线的精细紧致雅可比簇的分类

On the classification of fine compactified Jacobians of nodal curves

本文引入藤型精细紧致雅可比簇，刻画其作为光滑曲线雅可比簇极限及Néron模型紧致化的条件，并给出轨道偏序集的组合刻画。

Newton-Puiseux Roots of Jacobian Determinants

Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=γ(y)$, of the Jacobian $J=f_yg_x-f_xg_y$. We define the tree-model, $T(f,g)$, for the pair, using the contact orders of the Newton-Puiseux roots of $f$ and $g$. We then describe how the $γ$'s climb, and where they leave, the tree. We shall also show by two examples that the way the $γ$'s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one …

节点曲线可光滑化紧致 Jacobian 的完整理论

A complete theory of smoothable compactified Jacobians of nodal curves

本文引入并研究了节点曲线的 V-紧致 Jacobian，证明其严格大于经典紧致 Jacobian，并将其表征为单参数光滑化下光滑曲线 Jacobian 的极限，推广了先前关于精细紧致 Jacobian 的工作。

节点曲线上线丛的稳定性条件

Stability conditions for line bundles on nodal curves

本文引入了节点曲线上线丛及其退化的组合稳定性条件，证明了可光滑化精细紧致雅可比簇与这些条件存在双射，并证明了精细紧致泛雅可比簇均同构于Caporaso、Pandharipande和Simpson的构造。