Exceptional collections, and the Néron-Severi lattice for surfaces
type: Item
instance of: scholarly article
title: Exceptional collections, and the Néron-Severi lattice for surfaces
zbMATH Open document ID: 1387.14061
author: Charles Vial
published in: Advances in Mathematics
publication date: 2016-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.01776
reviewed by: Julien Grivaux
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14G27
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 6652656
zbMATH Keywords: derived category of coherent sheaves
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: rationality
zbMATH Keywords: exceptional collections
zbMATH Keywords: motives
zbMATH Keywords: projective space
MaRDI profile type: Publication
OpenAlex ID: W1921920143
arXiv ID: 1504.01776
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DOI: 10.1016/J.AIM.2016.10.012
review text: Let \(X\) be a smooth projective surface over a field \(k\), which can be chosen of any characteristic. In the present paper, the author tries to investigate when \(X\) has full or numerical exceptional collections of objects of maximal length. Recall that an object \(E\) of \(\mathrm{D}^{\mathrm{b}}(X)\) is exceptional if NEWLINE\[NEWLINE \mathrm{Hom}_{\mathrm{D}^{\mathrm{b}}(X)}(E, E[\ell]) = \begin{cases} k \text{ if }\ell=0\\ 0\text{ if }\ell \neq 0\end{cases}NEWLINE\]NEWLINE A family \(\{E_1, \dots, E_n \}\) of exceptional objects is an exceptional collection if and only if NEWLINE\[NEWLINE \mathrm{Hom}_{\mathrm{D}^{\mathrm{b}}(X)}(E_i, E_j[\ell]) =0\text{ if }i>j\text{ and }\ell \in \mathbb{Z}.NEWLINE\]NEWLINE This collection is full if it generates the category \(\mathrm{D}^{\mathrm{b}}(X)\). The most celebrated example of a full exceptional collection is due to Beilinson: \(\{O, O(1), \ldots, O(n)\}\) is a full exceptional collection on \(\mathbb{P}^n\). These definitions can be weakened as follows: an object \(E\) of \(\mathrm{D}^{\mathrm{b}}(X)\) is numerically exceptional if \(\chi(E, E)=1\). A family \(\{E_1, \dots, E_n \}\) of numerically exceptional objects is a numerically exceptional collection if and only if \(\chi(E_i, E_j) =0\) if \(i>j\). The collection is full if it generates the numerical Grothendieck group of \(X\). If \(X\) is a surface such that \(\chi(\mathcal{O}_X)=1\), the main result of the paper is a necessary and sufficient condition on the Neron-Severi lattice of \(X\) that ensures that \(X\) admits a full numerically exceptional collection. Some consequences of this results are presented.
description: scientific article; zbMATH DE number 6652656