Lotuses as computational architectures

TL;DR

This paper introduces a novel computational framework using lotus structures, derived from complex plane curve singularities, to systematically compute various invariants and properties of singularities.

Contribution

It establishes a new method to associate lotus structures with constellations of crosses, enabling systematic computation of invariants of plane curve singularities.

Findings

Lotus structures can encode invariants like log-discrepancies and Milnor numbers.

The method applies to complex singularities and adapts to positive characteristic.

Provides examples illustrating the computational process.

Abstract

Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve singularity by a sequence of toroidal modifications with respect to suitable local coordinates, one gets a naturally associated lotus, which allows to unify the classical trees used to encode the combinatorial type of the singularity. In this paper we explain how to associate a lotus to each constellation of crosses, which is a finite constellation of infinitely near points endowed with compatible germs of normal crossings divisors with two components, and how this lotus may be seen as a computational architecture. Namely, if the constellation of crosses is associated to an embedded resolution of a complex reduced plane curve singularity $A$, one may compute progressively as vertex and edge weights on the lotus the log-discrepancies of the exceptional divisors, the orders of vanishing on them of the starting coordinates, the multiplicities of the strict transforms of the branches of $A$, the orders of vanishing of a defining function of $A$, the associated Eggers-Wall tree, the delta invariant and the Milnor number of $A$, etc. We illustrate these computations using three recurrent examples. Finally, we describe the changes to be done when one works in positive characteristic.