发表机构
Instytut Matematyczny, Polska Akademia Nauk; Universidade de São Paulo, ICMC（波兰科学院数学研究所; 圣保罗大学计算与数学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限全纯映射下超曲面奇点的数值不变量，证明Segre数及一般Lê数的单调性，并给出Milnor数不等式的代数证明。
AI 中文摘要
我们研究了光滑胚之间有限全纯映射下超曲面奇点的数值比较。我们保留了Jelonek在孤立情形下关于Milnor数不等式的拓扑证明，并利用循环分解、Brieskorn模和迹给出了代数证明。对于约化除子理论拉回，我们证明了Segre数的字典序单调性。当两个奇点轨迹具有相同维数且目标Jacobian理想为等倍时，分量单调性成立。对于一维奇点轨迹，我们还在第一个非零Segre数相等或非分歧条件下获得了分量比较。论证使用了Segre环的有限平坦拉回和Hilbert–Samuel重数。通过与一般Lê数的等同，这些结果同样适用于这些不变量。我们将这些陈述与任意预极坐标中的比较区分开来，并将剩余的分量情形和分歧情形表述为猜想。
英文摘要
We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert--Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.
