Vol. 296, No. 1, 2018

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ISSN: 0030-8730
Monads on projective varieties

Simone Marchesi, Pedro Macias Marques and Helena Soares

Vol. 296 (2018), No. 1, 155–180
Abstract

We generalize Fløystad’s theorem on the existence of monads on projective space to a larger set of projective varieties. We consider a variety X, a line bundle L on X, and a basepoint-free linear system of sections of L giving a morphism to projective space whose image is either arithmetically Cohen–Macaulay (ACM) or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers a, b and c for a monad of type

0 (L)a O Xb Lc 0

to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterize low-rank vector bundles that are the cohomology sheaf of some monad as above.

Finally, we obtain an irreducible family of monads over projective space and make a description on how the same method could be used on an ACM smooth projective variety X. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional X and show that in one case this moduli space is irreducible.

Keywords
monads, ACM varieties
Mathematical Subject Classification 2010
Primary: 14F05
Secondary: 14J10, 14J60
Milestones
Received: 19 November 2016
Revised: 30 December 2017
Accepted: 30 December 2017
Published: 1 May 2018
Authors
Simone Marchesi
Instituto de Matemática, Estatística e Computação Científica
Universidade Estadual de Campinas
Cidade Universitária “Zeferino Vaz”
Campinas
Brazil
Pedro Macias Marques
Departamento de Matemática
Escola de Ciências e Tecnologia
Centro de Investigação em Matemática e Aplicações
Instituto de Investigação e Formação Avançada
Universidade de Évora
Évora
Portugal
Helena Soares
Departamento de Matemática
Instituto Universitário de Lisboa (ISCTE-IUL)
UNIDE (BRU-Business Research Unit)
Lisboa
Portugal