线排列的Bourbaki度

中文摘要 AI 辅助

我们研究了射影平面中线排列的Bourbaki度及其与相交格和梯度理想的合冲模的相互作用。我们证明了Bourbaki度可通过在第一个合冲模的初始度处评估约化特征多项式而获得。这导出了一个尖锐的上界以及涉及两个最大相交重数的细化结果，对全局Tjurina数、自由缺陷和Terao猜想产生了影响，包括当较小指数至少为5时对Dimca数值判据的改进。我们还建立了加法-删除公式，并证明了在添加直线时，与尖锐界之间的缺陷是单调的。最后，我们证明了对于至多八条直线的排列，Bourbaki度由相交格决定，而对于至少九条直线的每一种数量，都存在具有同构相交格但不同Bourbaki度的例子。

英文摘要

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition--deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.

发表机构

• Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究所）
• Universidade Federal de São Paulo (UNIFESP)（圣保罗联邦大学）

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