Local resolution of the second log-Fitting ideal

Abstract

Logarithmic Fitting ideals encode the complexity of the corresponding morphisms in a natural and intrinsic way, and their principalisation offers a new perspective on problems such as the resolution of cotangent sheaves and the monomialisation of morphisms. In this paper, we give a local resolution theorem for log-rank adapted morphisms: after finite sequences of smooth local blow-ups, the first two log-Fitting ideals \mathcal{F}\_{n--1}(σ) and \mathcal{F}\_{n--2}(σ) become principal monomial ideals.