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Strict germs on normal surface singularities

Matteo Ruggiero

TL;DR

The paper studies strict holomorphic germs between normal surface singularities of topological degree $1$, proving that any such germ factors as $f=\\pi\\circ\\sigma$ with $\\pi$ a modification and $\\sigma$ a local isomorphism (a Kato germ). It provides two proofs of this decomposition: a local flattening approach and a valuation-theoretic method, and then uses these ideas to relate strict selfmaps to sandwiched singularities via Kato surfaces. A key result is that strict selfmaps occur precisely on sandwiched singularities, and the eigenvaluation associated to a strict selfmap cannot be divisorial; the work also develops the valuative dynamics on $\\mathcal{V}_X$ to control the orbits and deduce structural consequences. These methods connect birational geometry, non-Archimedean dynamics, and the geometry of Kato surfaces to classify selfsimilar normal surface singularities and to illuminate Nash-type resolution strategies.

Abstract

We show that any holomorphic germ $f \colon (X,x_0) \to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=π\circ σ$, where $π\colon Y' \to (Y,y_0)$ is a modification and $σ\colon (X,x_0) \to (Y',y_1)$ is a local isomorphism sending $x_0$ to a point $y_1 \in π^{-1}(y_0)$. A result by Fantini, Favre and myself guarantees that when $f$ is a selfmap, then $(X,x_0)$ is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.

Strict germs on normal surface singularities

TL;DR

The paper studies strict holomorphic germs between normal surface singularities of topological degree , proving that any such germ factors as with a modification and a local isomorphism (a Kato germ). It provides two proofs of this decomposition: a local flattening approach and a valuation-theoretic method, and then uses these ideas to relate strict selfmaps to sandwiched singularities via Kato surfaces. A key result is that strict selfmaps occur precisely on sandwiched singularities, and the eigenvaluation associated to a strict selfmap cannot be divisorial; the work also develops the valuative dynamics on to control the orbits and deduce structural consequences. These methods connect birational geometry, non-Archimedean dynamics, and the geometry of Kato surfaces to classify selfsimilar normal surface singularities and to illuminate Nash-type resolution strategies.

Abstract

We show that any holomorphic germ of topological degree between normal surface singularities can be written as , where is a modification and is a local isomorphism sending to a point . A result by Fantini, Favre and myself guarantees that when is a selfmap, then is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.
Paper Structure (32 sections, 17 theorems, 32 equations, 2 figures)

This paper contains 32 sections, 17 theorems, 32 equations, 2 figures.

Table of Contents

  1. Basics on valuation spaces
  2. Valuation spaces
  3. Types of valuations
  4. Divisorial valuations
  5. Quasimonomial valuations
  6. Curve semivaluations
  7. Infinitely singular valuations
  8. Center of a valuation
  9. Valuation spaces as projective limits of dual graphs
  10. Map induced on valuation spaces
  11. Definitions and first properties
  12. Geometric interpretation of the action
  13. Detecting indeterminacy points
  14. Log-discrepancy
  15. Jacobian formula
  16. ...and 17 more sections

Key Result

Theorem A

Let $f\colon (X,x_0) \to (Y,y_0)$ be a strict germ between two normal surface singularities. Then there exists a modification ${\pi}\colon Y' \to (Y,y_0)$, a point $y_1 \in {\pi}^{-1}(y_0)$, and a local isomorphism ${\sigma} \colon (X,x_0) \to (Y',y_1)$ so that $f={\pi} \circ {\sigma}$.

Figures (2)

  • Figure 1: Example of a Kato datum on the quotient singularity of \ref{['ssec:strictnoeff']} seen as a sandwiched singularity. Self-intersections on curves are indicated for $Y'$, they are omitted for ($-1$)-curves.
  • Figure 2: Kato datum over the quotient singularity of type $\frac{1}{5k-3}(1,2k-1)$. The dashed curves are the critical curves and their strict transforms. On the right side, the associated Kato surface $S$. Self-intersections on curves are indicated for $Z$, $Z'$ and $S$, they are omitted for ($-1$)-curves.

Theorems & Definitions (48)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2: favre-jonsson:eigenval
  • Proposition 2.3: favre-jonsson:eigenval
  • ...and 38 more