On the jumping lines of bundles of logarithmic vector fields along plane curves

IF 0.8 3区 数学 Q2 MATHEMATICS
A. Dimca, Gabriel Sticlaru
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引用次数: 8

Abstract

For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$, we study the set of jumping lines for the rank two vector bundle $T\langle C \rangle $ on $\mathbb{P}^2$, whose sections are the logarithmic vector fields along $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T\langle C \rangle $ is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourbaki ideal, a result going back to Schwarzenberger. Other classical general results by Barth, Hartshorne and Hulek resurface in the study of this special class of rank two vector bundles.
沿平面曲线的对数向量场束的跳线
对于复射影平面$\mathbb{P}^2$上的简化曲线$C:f=0$,研究了$\mathbb{P}^2$上的二阶向量束$T\langle C \rangle $的跳线集,其截面是沿$C$的对数向量场。指出了这些跳线与$f$的雅可比矩阵模的Lefschetz型性质和$f$的雅可比合集模的Bourbaki理想的关系。特别地,当向量束T\langle C \rangle $是不稳定的,一条线是跳线当且仅当它满足由布尔巴基理想定义的0维子格式,这个结果可以追溯到施瓦岑贝格。Barth, Hartshorne和Hulek的其他经典一般结果在对这类特殊的二阶向量束的研究中重新出现。
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来源期刊
CiteScore
1.60
自引率
0.00%
发文量
29
审稿时长
>12 weeks
期刊介绍: Publicacions Matemàtiques is a research mathematical journal published by the Department of Mathematics of the Universitat Autònoma de Barcelona since 1976 (before 1988 named Publicacions de la Secció de Matemàtiques, ISSN: 0210-2978 print, 2014-4369 online). Two issues, constituting a single volume, are published each year. The journal has a large circulation being received by more than two hundred libraries all over the world. It is indexed by Mathematical Reviews, Zentralblatt Math., Science Citation Index, SciSearch®, ISI Alerting Services, COMPUMATH Citation Index®, and it participates in the Euclid Project and JSTOR. Free access is provided to all published papers through the web page. Publicacions Matemàtiques is a non-profit university journal which gives special attention to the authors during the whole editorial process. In 2019, the average time between the reception of a paper and its publication was twenty-two months, and the average time between the acceptance of a paper and its publication was fifteen months. The journal keeps on receiving a large number of submissions, so the authors should be warned that currently only articles with excellent reports can be accepted.
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