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$\mathbb{G}_m$-Equivariant Degenerations of del Pezzo Surfaces

Junyao Peng

TL;DR

The work advances the understanding of $\\mathbb{G}_m$-equivariant degenerations of del Pezzo surfaces by linking special valuations to degenerations via dual complexes and quasi-monomial valuations. It proves the global structural results: the space of special valuations is connected and decomposes into a locally finite partition into intervals, each producing an isomorphic family of degenerations, with a local cone structure near divisorial centers. The approach blends valuation theory, MMP techniques, and qdlt Fano-type models to realize higher-rank degenerations of log Fano varieties, confirming a global analogue of Kollár-type conjectures in this setting and providing detailed local models and degeneration diagrams. The paper also offers concrete examples and a weaker local result for del Pezzo surfaces with quotient singularities, illustrating the geometry of special degenerations and their dual complexes.

Abstract

We study special $\mathbb{G}_m$-equivariant degenerations of a smooth del Pezzo surface $X$ induced by valuations that are log canonical places of $(X,C)$ for a nodal anti-canonical curve $C$. We show that the space of special valuations in the dual complex of $(X,C)$ is connected and admits a locally finite partition into sub-intervals, each associated to a $\mathbb{G}_m$-equivariant degeneration of $X$. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.

$\mathbb{G}_m$-Equivariant Degenerations of del Pezzo Surfaces

TL;DR

The work advances the understanding of -equivariant degenerations of del Pezzo surfaces by linking special valuations to degenerations via dual complexes and quasi-monomial valuations. It proves the global structural results: the space of special valuations is connected and decomposes into a locally finite partition into intervals, each producing an isomorphic family of degenerations, with a local cone structure near divisorial centers. The approach blends valuation theory, MMP techniques, and qdlt Fano-type models to realize higher-rank degenerations of log Fano varieties, confirming a global analogue of Kollár-type conjectures in this setting and providing detailed local models and degeneration diagrams. The paper also offers concrete examples and a weaker local result for del Pezzo surfaces with quotient singularities, illustrating the geometry of special degenerations and their dual complexes.

Abstract

We study special -equivariant degenerations of a smooth del Pezzo surface induced by valuations that are log canonical places of for a nodal anti-canonical curve . We show that the space of special valuations in the dual complex of is connected and admits a locally finite partition into sub-intervals, each associated to a -equivariant degeneration of . This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.

Paper Structure

This paper contains 20 sections, 34 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted blow-ups at a point
  4. Dual complexes
  5. Test configurations
  6. Quasi-monomial valuations
  7. Unicuspidal curves
  8. Special Valuations over Log Calabi-Yau Surface Pairs
  9. Numerical criterion for special valuations
  10. Special divisors over $X$
  11. Special valuations over $X$, Part I
  12. Special valuations over $X$, Part II
  13. Proof of Theorem \ref{['mainthm:classification_of_special_valuations_on_del_pezzo_surfaces']} and \ref{['mainthm:structure_of_the_space_of_special_valuations']}
  14. Qdlt Fano Type Models of Special Degenerations
  15. Notations and outline of the proof
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $X$ be a smooth del Pezzo surface of degree $d$. Let $C\sim -K_X$ be a normal crossing divisor on $X$. Suppose $C$ is nodal and $x\in C$ is a node. Let $v\in \text{QM}(X,C)$ be a quasi-monomial valuation over $X$.

Theorems & Definitions (86)

  • Conjecture 1.2: cf. LX24
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 76 more