德尔佩佐三折的分类托雷利定理新视角

IF 2.1 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees Pub Date : 2024-11-01 DOI:10.1016/j.matpur.2024.103627

Soheyla Feyzbakhsh , Zhiyu Liu , Shizhuo Zhang

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引用次数: 0

摘要

设 Yd 是皮卡等级为 1、度数为 d≥2 的德尔佩佐三褶。在本文中，我们通过 Yd 有界派生类的一个特殊可容许子类（称为库兹涅佐夫分量），运用两种不同的观点来研究 Yd：（i）布里尔-诺ether 重构。我们证明，Yd 可以唯一地复原为其库兹涅佐夫成分中布里奇兰稳定对象的布里奇兰-诺ether 位置。我们证明，只要与明确的自等价构成，两个阶数为 2≤d≤4 的 del Pezzo 三维库兹涅佐夫分量的任何傅立叶-穆凯类型等价都可以提升为它们有界派生范畴的等价。因此，我们得到了 Yd 的库兹涅佐夫分量的傅里叶-穆凯类型自等价群的完整描述。在附录中，我们对四元双实体上的瞬子剪进行了分类，推广了德鲁尔的一个结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let

Y_{d}

be a del Pezzo threefold of Picard rank one and degree

d \geq 2

. In this paper, we apply two different viewpoints to study

Y_{d}

via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component:

(i)
Brill–Noether reconstruction. We show that $Y_{d}$ can be uniquely recovered as a Brill–Noether locus of Bridgeland stable objects in its Kuznetsov component.
(ii)
Exact equivalences. We prove that up to composing with an explicit auto-equivalence, any Fourier–Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree $2 \leq d \leq 4$ can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier–Mukai type auto-equivalences of the Kuznetsov component of $Y_{d}$ .

In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.

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来源期刊

Journal de Mathematiques Pures et Appliquees 数学-数学

CiteScore

4.30

自引率

0.00%

发文量

审稿时长

6 months

期刊介绍： Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.