Monads on projective varieties
Simone Marchesi, Pedro Macias Marques, and Helena Soares

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 , a line bundle on , and a base-point-free linear system of sections of 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 , , and 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…
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