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Birational geometry of Calabi-Yau pairs $(\mathbb{P}^3, D)$ of coregularity 2

Eduardo Alves da Silva

Abstract

This paper aims to study the birational geometry of log Calabi-Yau pairs$(\mathbb{P}^3, D)$ of coregularity 2, where in this case $D$ is an irreducible normal quartic surface with canonical singularities. We completely classify which toric weighted blowups of a point will initiate a volume preserving Sarkisov link starting with this pair. Depending on the type of singularity, our results point out that some of these weights do not work generically for a general member of the corresponding coarse moduli space of quartics.

Birational geometry of Calabi-Yau pairs $(\mathbb{P}^3, D)$ of coregularity 2

Abstract

This paper aims to study the birational geometry of log Calabi-Yau pairs of coregularity 2, where in this case is an irreducible normal quartic surface with canonical singularities. We completely classify which toric weighted blowups of a point will initiate a volume preserving Sarkisov link starting with this pair. Depending on the type of singularity, our results point out that some of these weights do not work generically for a general member of the corresponding coarse moduli space of quartics.
Paper Structure (36 sections, 16 theorems, 42 equations, 4 figures, 8 tables)

This paper contains 36 sections, 16 theorems, 42 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Outline of the proofs of the Theorems \ref{['thm vp weights']} & \ref{['thm vp weights sark']}.
  3. Structure of the paper
  4. Acknowledgements
  5. Overview about Calabi-Yau pairs and the Sarkisov Program
  6. Sarkisov Program.
  7. The coregularity
  8. A short compendium on singularities of quartics surfaces
  9. Explicit resolution of Du Val singularities
  10. Volume preserving Sarkisov links
  11. First approach to the problem:
  12. Determination of the volume preserving weights
  13. Notation for $D \subset \mathbb{P}^3$ quartic surface and some generalities
  14. The $A_n$ case
  15. Criteria for $A_n$ singularities on a quartic surface
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $(\mathbb{P}^3,D)$ be a log Calabi-Yau pair of coregularity 2 and $\pi \colon (X,D_X) \rightarrow (\mathbb{P}^3,D)$ be a volume preserving toric $(1,a,b)$-weighted blowup of a torus invariant point. Then this point is necessarily a singularity of $D$ and, up to permutation, the only possibilitie

Figures (4)

  • Figure 1: Resolution of the singularity $A_n$ for $n$ even.
  • Figure 2: Fan of the $(1,a,b)$-weighted blowup of $\mathbb{P}^3$.
  • Figure 3: Case $a+1 \leq j \leq n-a < b$.
  • Figure 4: Case $j=n-a+1$.

Theorems & Definitions (35)

  • Theorem 1.1: See Theorem \ref{['thm vp weights']}
  • Theorem 1.2: See Theorem \ref{['thm vp weights sark']}
  • Remark 1.3: Important
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Proposition 2.5: cf. ck Remark 1.7
  • Remark 2.6
  • Definition 2.7
  • ...and 25 more