Frobenius splitting and derived category of toric varieties
type: Item
instance of: scholarly article
title: Frobenius splitting and derived category of toric varieties
zbMATH Open document ID: 1230.14018
author: Laura Costa
author: Rosa M. Miró-Roig
published in: Illinois Journal of Mathematics
publication date: 2011-11-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.5315
full work available at URL: https://projecteuclid.org/euclid.ijm/1318598676
review text: Let \(X\) be a smooth complete toric variety over an algebraically closed field of characteristic \(0\). \textit{A. King} [``Tilting bundles on some rational surfaces'', \url{http://www.maths.bath.ac.uk/~masadk/papers/tilt.pdf}] conjectured that \(X\) has a full strongly exceptional collection of line bundles, that is, that the bounded derived category \(D(X)\) of complexes of coherent sheaves on \(X\) admits a semiorthogonal decomposition by exceptional line bundles with extra orthogonality properties. A counterexample to this conjecture is provided by \textit{L. Hille} and \textit{M. Perling} [Compos. Math. 142, No. 6, 1507--1521 (2006; Zbl 1108.14040)]: for a blow up of three points in the degree 2 Hirzebruch surface, the exceptional collections fail to be strong. The authors notice that the counterexample is not a Fano variety and then they conjecture King's statement to hold for any smooth complete Fano toric variety over an algebraically closed field of characteristic \(0\). In this paper, they prove that King's conjecture holds indeed for such varieties of (almost) maximal Picard number: if \(X\) is \(n\)-dimensional, and \(n\) odd, for \(\rho_X = 2n-1\), if \(X\) is \(n\)-dimensional, and \(n\) even, for \(\rho_X = 2n\) and \(\rho_X = 2n-1\). The main ingredients in the proof are the classification of smooth complete Fano toric varieties with such Picard numbers, and a case-by-case analysis of the splitting of \((\pi_p)_*({\mathcal O}_X)\), where \(\pi_p\) is the Frobenius morphism.
reviewed by: Marcello Bernardara
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 5981647
zbMATH Keywords: toric varieties
zbMATH Keywords: King's conjecture
zbMATH Keywords: exceptional collection
zbMATH Keywords: Frobenius splitting
MaRDI profile type: Publication
arXiv ID: 1006.5315
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description: scientific article