Finite holomorphic pullbacks, Segre numbers, and generic L\^e numbers

Abstract

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 L\^e 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.