Monads on projective varieties

Simone Marchesi, Pedro Macias Marques and Helena Soares

Vol. 296 (2018), No. 1, 155–180

DOI: 10.2140/pjm.2018.296.155

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

$$0\to {\left(L^{\vee }\right)}^a\to {\mathcal{O}}_X^b\to L^c\to 0$$

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