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Fourier--Mukai partners of elliptic ruled surfaces over arbitrary characteristic fields

Hokuto Uehara, Tomonobu Watanabe

Abstract

The first author explicitly describes the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field in \cite{Ue17}. In this article, we generalize it over arbitrary characteristic fields. We also obtain a partial evidence of the Popa--Schnell conjecture in the proof.

Fourier--Mukai partners of elliptic ruled surfaces over arbitrary characteristic fields

Abstract

The first author explicitly describes the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field in \cite{Ue17}. In this article, we generalize it over arbitrary characteristic fields. We also obtain a partial evidence of the Popa--Schnell conjecture in the proof.
Paper Structure (14 sections, 19 theorems, 106 equations)

This paper contains 14 sections, 19 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Acknowledgements
  4. Relative moduli spaces of sheaves on elliptic fibrations
  5. Fourier--Mukai partners of elliptic surfaces
  6. Weil--Châtelet group
  7. Elliptic curves and automorphisms
  8. Pirozhkov's result and its application
  9. Fourier--Mukai partners of elliptic ruled surfaces
  10. Singular fibers of elliptic ruled surfaces
  11. Case (i-2).
  12. Proof of Theorem \ref{['thm:main']}.
  13. Case: $m=p^e\ge 5$ for some $e>0$.
  14. Case: Arbitrary $m\ge 5$ with $m\ne p^e$ for any $e>0$.

Key Result

Theorem 1.1

Let $S$ be an elliptic ruled surface defined over $k$ and $\pi \colon S \to E$ be a $\mathbb P^1$-bundle over an elliptic curve $E$. If $|\mathop{\mathrm{FM}}\nolimits (S)|\ne 1$, then $S$ is of the form for some $\mathcal{L} \in \mathop{\mathrm{Pic}}\nolimits^0{E}$ of order $m \ge 5$. Furthermore we have and Here, $\varphi$ is the Euler function, and we define as a subgroup of $(\mathbb{Z}/m\

Theorems & Definitions (47)

  • Theorem 1.1
  • Proposition 1.2: =Corollary \ref{['corollary:base-elliptic']}
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 37 more