On the integral variation map of isolated plane curves singularities

TL;DR

This paper develops an explicit, algorithmic approach to compute the integral variation map and algebraic monodromy of isolated plane curve singularities, using geometric bases, gyrographs, and a new combinatorial framework, with implementation in Python.

Contribution

It introduces a novel algorithmic method and combinatorial structures for computing monodromy and variation maps of plane curve singularities, with explicit bases and software implementation.

Findings

Explicit matrices for monodromy and variation maps are computed for any Milnor fiber.

Gyrographs encode monodromy and variation maps via geometric and combinatorial data.

The methods are implemented in a publicly available Python program.

Abstract

The integral variation map and algebraic monodromy of isolated plane curve singularities are important homological invariants of the singularity which are still far from being completely understood. This work provides effective ways of computing them with respect to an explicit geometric basis of the homology. For any given topological type of plane curve singularity, we construct an analytic model of it, along with a vector field on our version of its A'Campo space. This vector field is tangent to the Milnor fibers at radius zero and the union of the stable manifolds of their singularities yields a spine of each fiber, which can be described explicitly. This is very much inspired by a recent work of the authors. Our first main contribution is the algorithmic computation of the algebraic monodromy and integral variation map as matrices with explicit bases for any Milnor fiber in the Milnor fibration, not merely congruence classes. For our second contribution, we introduce gyrographs which are graphs equipped with angular data and rational weights. We prove that the invariant spine naturally carries a gyrograph structure were the weights are given by the Hironaka numbers, and that this structure recovers the geometric monodromy as a homotopy class, as well as the integral variation map. This provides a combinatorial framework for computation by hand. Our methods are further implemented in a publicly availablecomputer program written in Python.