具有若干预设奇点的二次曲线-直线构形的拓扑不等式
Topological inequalities for conic-line arrangements with some prescribed singularities
- University of the National Education Commission（国家教育委员会大学）
- Aix-Marseille Université（艾克斯-马赛大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究从4维流形视角引入组合二次曲线-直线构形，证明了ADE型偶总次数构形的Hirzebruch型不等式，还通过相关理论得到可分与奇数型构形消解重数序列的不等式与同余式。
AI中文摘要：
我们从4维流形的视角研究复射影平面中的二次曲线与直线构形。我们引入组合二次曲线-直线构形，它是秩3拟阵的二次曲线-直线对应物，以及它们的拓扑与光滑实现，并探究当复构形的分量被替换为同调类相同、局部奇点模型相同的局部平坦球面时，哪些限制条件仍然成立。对于具有拓扑实现的偶总次数的ADE型组合二次曲线-直线构形，我们证明了一个Hirzebruch型不等式；该证明仅使用分支覆盖和有理二重点的局部拓扑。在复代数范畴中，我们证明当且仅当该构形是最大化曲线时等号成立，因此它是具有预设指数的自由构形。随后，我们引入可分与奇数型组合二次曲线-直线构形，并通过循环分支覆盖、G-符号定理、旋结构以及Furuta的10/8定理，得到了它们的消解重数序列的不等式与同余式。
英文摘要：
We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models. For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.
