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Monads on projective varieties
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We generalize Fløystad's theorem on the existence of monads on projectivespace to a larger set of projective varieties. We consider a varietyX, a linebundleLonX, and a basepoint-free linear system of sections ofLgiving amorphism to projective space whose image is either arithmetically Cohen-Macaulay (ACM) or linearly normal and not contained in a quadric. Wegive necessary and sufficient conditions on integersa,bandcfor a monadof type $\mathbf{0} \rightarrow\left(\boldsymbol{L}^{\vee}\right)^{a} \rightarrow \mathcal{O}_{X}^{b} \rightarrow \boldsymbol{L}^{c} \rightarrow \mathbf{0}$ to exist. We show that under certain conditions there exists a monad whosecohomology sheaf is simple. We furthermore characterize low-rank vectorbundles that are the cohomology sheaf of some monad as above.Finally, we obtain an irreducible family of monads over projective spaceand make a description on how the same method could be used on an ACMsmooth projective varietyX. We establish the existence of a coarse modulispace of low-rank vector bundles over an odd-dimensionalXand show thatin one case this moduli space is irreducible.
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MARCHESI, Simone, MARQUES, Pedro Macías and SOARES, Helena. Monads on projective varieties. Pacific Journal of Mathematics. 2018. Vol. 296, num. 1, pags. 155-180. ISSN 0030-8730. [consulted: 19 of August of 2026]. Available at: https://hdl.handle.net/2445/142819