Derived categories of Fano threefolds and degenerations

• 成果类型: Article
• 署名作者: Kuznetsov, Alexander; Shinder, Evgeny
• 署名单位: Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; HSE University (National Research University Higher School of Economics); University of Sheffield; University of Bonn
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01304-x
• 发表日期: 2025
• 页码: 377-430
• 关键词: DECOMPOSITIONS; SINGULARITIES

摘要

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree d is an element of{2,3,4,5}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d \in \{2,3,4,5\}$\end{document} and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d = 1$\end{document} can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus g=2d+2 is an element of{4,6,8,10,12}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g = 2d + 2 \in \{4,6,8,10,12\}$\end{document}. This corrects and proves the Fano threefolds conjecture of the first author from (Kuznetsov in Tr. Mat. Inst. Steklova 264:116-128, 2009), and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds.