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On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective

Eduardo Alves da Silva

Abstract

This paper aims to study the decomposition group of a nonsingular plane cubic under the light of the log Calabi-Yau geometry. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of this group is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. We also negatively answer a question posed by Blanc, Pan and Vust asking whether the canonical complex of a nonsingular plane cubic is split. Within a similar context em dimension 3, we exhibit in detail an interesting counterexample for a possible generalization of a theorem by Pan in which there exists a Sarkisov factorization obtained algorithmically that is not volume preserving.

On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective

Abstract

This paper aims to study the decomposition group of a nonsingular plane cubic under the light of the log Calabi-Yau geometry. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of this group is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. We also negatively answer a question posed by Blanc, Pan and Vust asking whether the canonical complex of a nonsingular plane cubic is split. Within a similar context em dimension 3, we exhibit in detail an interesting counterexample for a possible generalization of a theorem by Pan in which there exists a Sarkisov factorization obtained algorithmically that is not volume preserving.
Paper Structure (21 sections, 14 theorems, 10 equations, 16 figures)

This paper contains 21 sections, 14 theorems, 10 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgements
  4. Log Calabi-Yau geometry and the volume preserving Sarkisov Program
  5. Sarkisov Program.
  6. Geometry of the Hirzebruch surfaces $\mathbb{F}_n$.
  7. The decomposition group of a nonsingular plane cubic
  8. Decomposition and inertia groups.
  9. Sarkisov link of type I:
  10. Sarkisov link of type II:
  11. Sarkisov link of type III:
  12. Sarkisov link of type IV:
  13. The canonical complex of $C \subset \mathbb{P}^2$
  14. Question by Blanc, Pan & Vust.
  15. Volume preserving x standard Sarkisov factorization
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $C \subset \mathbb{P}^2$ be a nonsingular cubic. The standard Sarkisov Program applied to an element of $\mathop{\mathrm{Dec}}\nolimits(C)$ is automatically volume preserving.

Figures (16)

  • Figure 2: Element of $\mathop{\mathrm{Dec}}\nolimits(C)$.
  • Figure 3: Step $i$ of the Sarkisov Program.
  • Figure 4: Case 1 - Sarkisov link of type II.
  • Figure 5: Case 2 - Sarkisov link of type II.
  • Figure 6: Case 3 - Sarkisov link of type II.
  • ...and 11 more figures

Theorems & Definitions (34)

  • Theorem 1.1: See Theorem \ref{['thm sark=vp cubic']}
  • Theorem 1.2: See Theorem \ref{['non split can comp']}
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 2.2: cf. ck Remark 1.7
  • Remark 2.3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 24 more