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Semistable degenerations of double octics

Marcin Oczko

Abstract

We present an algorithm for computing semistable degeneration of double octic Calabi-Yau threefolds. Our method has a combinatorial representation by the means of double octic diagrams. The proposed algorithm is applicable both in classical context over a complex disk as well as in arithmetic setting over a spectrum of DVR. We illustrate algorithm's efficacy through three examples where we compute semistable degeneration and limiting mixed Hodge structure for explicit families of double octics.

Semistable degenerations of double octics

Abstract

We present an algorithm for computing semistable degeneration of double octic Calabi-Yau threefolds. Our method has a combinatorial representation by the means of double octic diagrams. The proposed algorithm is applicable both in classical context over a complex disk as well as in arithmetic setting over a spectrum of DVR. We illustrate algorithm's efficacy through three examples where we compute semistable degeneration and limiting mixed Hodge structure for explicit families of double octics.
Paper Structure (34 sections, 9 theorems, 27 equations)

This paper contains 34 sections, 9 theorems, 27 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Infinitesimal deformations of double octics
  4. Octics diagrams
  5. New $l_3$ line
  6. New $p_{4}^{0}$ point
  7. A $p_{5}^{1}$ point degenerates to a $p_{5}^{2}$ point
  8. Two $p_{4}^{1}$ points collide to a $p_{5}^{2}$ point
  9. Two $p^1_4$ points collide to a $p^1_5$ point
  10. A $p_{4}^{0}$ point degenerates to a $p_{5}^{2}$ point
  11. New $p_{4}^{1}$ point
  12. A $p_{4}^{0}$ point degenerates to a $p_{4}^{1}$ point
  13. A $p_{4}^{0}$ point degenerates to a $p_{5}^{1}$ point
  14. A $p_{5}^{0}$ point degenerates to a $p_{5}^{2}$ point
  15. A $p_{5}^{0}$ point degenerates to a $p_{5}^{1}$ point
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $D \subset \mathbb P^3$ be an octic arrangement. Then there exists a sequence $\sigma = \sigma_1 \circ ... \circ \sigma_s \colon \widetilde{\mathbb P^3} \rightarrow \mathbb P^3$ of blowing-ups and a smooth and even divisor $D^* \subset \widetilde{\mathbb P^3}$ such that $\sigma_*(D^*) = D$ and t

Theorems & Definitions (13)

  • Theorem 2.1: CYB
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Definition 5.1
  • Theorem 5.2
  • Proposition 5.3
  • proof
  • Theorem 5.4: li
  • Proposition 5.5
  • ...and 3 more