#### Vol. 296, No. 1, 2018

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### 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

 $0\to {\left({L}^{\vee }\right)}^{a}\to {\mathsc{O}}_{X}^{b}\to {L}^{c}\to 0$

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.