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The total spine of the Milnor fibration of a plane curve singularity

Pablo Portilla Cuadrado, Baldur Sigurðsson

TL;DR

This work develops a geometric-convex framework to realize the vanishing topology of plane curve singularities via a spine inside each Milnor fiber. By leveraging the gradient flow of $-\nabla \log|f|$ and a carefully constructed real oriented blow-up together with polar-resolutions, the authors partition the separatrices into smooth strata and define an invariant Milnor fibration at radius zero. They prove that these strata determine a global spine formed by trajectories that do not escape the boundary, yielding a smooth, fiberwise spine compatible with the Milnor fibration and its radius-zero quotient. The results also provide detailed homogeneous-case analyses and lay groundwork for understanding integral monodromy and variation maps in this setting.

Abstract

For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field $-\nabla \log |f|$. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary.

The total spine of the Milnor fibration of a plane curve singularity

TL;DR

This work develops a geometric-convex framework to realize the vanishing topology of plane curve singularities via a spine inside each Milnor fiber. By leveraging the gradient flow of and a carefully constructed real oriented blow-up together with polar-resolutions, the authors partition the separatrices into smooth strata and define an invariant Milnor fibration at radius zero. They prove that these strata determine a global spine formed by trajectories that do not escape the boundary, yielding a smooth, fiberwise spine compatible with the Milnor fibration and its radius-zero quotient. The results also provide detailed homogeneous-case analyses and lay groundwork for understanding integral monodromy and variation maps in this setting.

Abstract

For any plane curve singularity defined by an analytic function germ , we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field . Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, , into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius . The strict transform of in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary.
Paper Structure (56 sections, 91 theorems, 439 equations, 41 figures)

This paper contains 56 sections, 91 theorems, 439 equations, 41 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Future work
  4. Organization of the paper
  5. General Setting
  6. The complex gradient as a lift
  7. The collapsing map
  8. Isometric coordinates
  9. Vector fields
  10. Embedded Resolutions
  11. Embedded resolutions
  12. Initial part and vanishing order
  13. Invariants of the resolution: $c_0, c_1$ and the canonical divisor
  14. Directions on the dual graph
  15. The tangent associated with a divisor
  16. ...and 41 more sections

Key Result

Theorem A

Assuming that $\mathbb{C}^2$ is endowed with a generic metric, the punctured total spine $S\setminus\{0\}$ is the disjoint union of strata, each of which is a punctured disk, an open solid torus or an open solid Klein bottle. Each stratum corresponds to a point $p$ on the exceptional divisor of an e

Figures (41)

  • Figure 2.2.0: On the left, the Milnor fiber of the singularity $y^2+x^3$. In blue we see the part of the Milnor fiber that maps to the origin by the collapsing map. In the central fiber we see the image of the boundary component of the Milnor fiber (red).
  • Figure 2.4.0: From left to right: a fountain, a sink and a saddle point.
  • Figure 2.4.2: A $4$-pronged singularity on the left corresponding to the vector field $\bar{z}^3$. And a $4$-pronged boundary on the right.
  • Figure 2.4.3: The vector field near the singularity of the vector field given by $z^3$: a $4$-petal singularity.
  • Figure 3.4.0: Effect of a blow-up on directed graphs. On top the effect of blowing up a smooth point at $D_j$. On bottom, the effect of blowing up the intersection point $D_j \cap D_k$.
  • ...and 36 more figures

Theorems & Definitions (284)

  • Theorem A: \ref{['thm:spine']}
  • Theorem B: \ref{['thm:invariant_spine']}
  • Remark 1
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 2
  • Remark 2
  • ...and 274 more