Monads on projective varieties

TL;DR

This paper extends the existence of monads from projective space to a broader class of projective varieties, characterizing when certain monads exist and describing their moduli spaces.

Contribution

It generalizes Fløystad's theorem to more varieties, providing necessary and sufficient conditions for monad existence and analyzing the moduli space of low-rank vector bundles.

Findings

Existence conditions for monads on various projective varieties.

Characterization of low-rank vector bundles as monad cohomology.

Construction of irreducible moduli spaces for certain vector bundles.

Abstract

We generalise Fl\o{}ystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the 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 \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the 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.