On the integral variation map of isolated plane curves singularities

TL;DR

The paper develops an explicit analytic-model framework and a combinatorial spine (A'Campo space) to compute integral monodromy and variation for isolated plane curve singularities. By constructing a chain-level description tied to the invariant spine and introducing gyrographs, it provides concrete, integer-valued matrices for monodromy and variation, not just congruence classes. The approach combines real oriented blow-ups, Seifert-fibered pieces, and detailed dynamical analysis of a horizontal vector field, yielding both theoretical invariants and practical hand/computational tools. A publicly available Python implementation enables algorithmic computation from a resolution graph all the way to homological invariants, enhancing the study and classification of plane curve singularities. The framework highlights the role of Hironaka numbers and offers a structured path from resolution data to explicit algebraic invariants, with validated examples including multi-Puiseux scenarios.

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.

Figures (11)

• Figure 2.1.1: The resolution graph of a plane curve with its directed structure. In green the branch $\Gamma[j]$ of $\Gamma$ at $j$ and in red, $\Gamma[i]$.
• Figure 2.1.2: A nice total order gives a natural embedding of the resolution graph on the real plane $\mathbb{R}^2$. The numbers on the arrows indicate the nice total order given on the set of branches as input data for a total order on the set of vertices and arrowheads as defined in \ref{['lem:total_order_vertices']}.
• Figure 3.3.1: On the left hand side we see, in red, the vertices that are in $S_{ik}$. On the right hand side we see: in red the vertices that contribute to $M_{ik}$; which together with those in blue are that the vertices that contribute to $M_{k}$.
• Figure 3.4.1: A representation of $\hat{D}_i$ corresponding to a vertex $i$ with four neighbor invariant vertices $j,k_1,k_2$ and $k_3$ (with $j \to i$); and one vertex in $\Gamma \setminus \Upsilon$ (in red). We can also see the two points of intersection, $p_1$ and $p_2$, of $\hat{D}_i$ with the strict transform of the generic polar curve. In this case $h_i(k_s) = s$ for $s=1,2,3$.
• Figure 3.4.2: Suppose that $i\to k$ is an edge, and that $m_i = 3$ and $m_k = 2$. We see the projection of $A^{\mathrm{inv}}_{ik}$ to $(\mathbb{R}/2\pi\mathbb{Z})^2$, with coordinates $(\tilde{\alpha}_k, \tilde{\beta}_k) = (\beta_{ik}, \alpha_i)$. The black subspace is the intersection with a Milnor fiber, given as a level set of $m_i \tilde{\alpha}_i + m_k \tilde{\beta}_k$. The geometric monodromy $G_k$ maps a point with coordinates $(\tilde{\alpha}_k, \tilde{\beta}_k)$ to a point with coordinates $(\tilde{\alpha}_k + 1/3, \tilde{\beta}_k)$, as indicated by the red arrow. Simiarly, $G_i$ adds $1/2$ to the $\alpha_i$ coordinate. If we fix some point $x \in (\mathbb{R}/2\pi\mathbb{Z})^2$, we have a segment in $\{x\}\times [0,1] \subset A^{\mathrm{inv}}_{ik}$. Projecting the image of this segment by $G_{ik} \to (\mathbb{R}/2\pi\mathbb{Z})^2$, we get the green segment in the picture, interpolating between $G_i$ and $G_k$.

Theorems & Definitions (54)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 5