A strictly decreasing invariant for resolution of singularities in dimension two

TL;DR

This paper introduces a new local invariant for resolving singularities in two-dimensional schemes, demonstrating its strict decrease throughout the resolution process, thus aiding in the systematic resolution of singularities.

Contribution

It constructs a strictly decreasing invariant specifically for 2D excellent Noetherian schemes with boundary, enhancing the resolution algorithm of Cossart, Jannsen, and Saito.

Findings

Invariant strictly decreases at each resolution step

Ensures progress in resolution process

Applicable to 2D excellent Noetherian schemes with boundary

Abstract

We construct a local invariant for resolution of singularities of 2-dimensional excellent Noetherian schemes with boundary. We prove that the invariant strictly decreases at every step of the algorithm of Cossart, Jannsen and Saito.