A polyhedral approach to the invariant of Bierstone and Milman

TL;DR

This work shows that the local resolution invariant of Bierstone and Milman in characteristic zero, $inv_X(x)$, can be recovered entirely from polyhedra associated to the singularity and their projections. By associating projected Newton-type polyhedra $\Delta(\mathcal{E},u,y)$ to pairs (and to history-augmented pairs), the authors link each step’s higher multiplicity entry $\nu_{r+1}(x)$ to a polyhedral invariant $\delta(\Delta(\mathcal{F}_r,u,y_r))$, corrected by the exceptional data. The paper develops a thorough framework: (i) a no-exceptional-divisor case where the invariant is read directly from polyhedra, (ii) an extension to pairs with history to handle exceptional divisors, and (iii) the full general case showing that $inv_X(x)$ is determined by polyhedral data across the resolution process. This polyhedral realization enables potential extensions to positive characteristic via weighted sections and provides a clear, constructive route to the BM invariant through combinatorial-geometric objects.

Abstract

Based on previous work by the author we deduce that the invariant introduced by Bierstone and Milman in order to give a proof for constructive resolution of singularities in characteristic zero can be determined purely by considering certain polyhedra and their projections.

Theorems & Definitions (42)

• Theorem 1: Theorem \ref{['Prop:ThmA2ndB']}
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Definition 2.7