Higher syzygy bundles and the Eisenbud-Huneke-Ulrich conjecture

Abstract: We study ideals in a polynomial ring with partially linear (virtual) resolutions. We establish an effective bound beyond which their powers coincide with a power of the maximal ideal, proving a slightly weaker version of the Eisenbud-Huneke-Ulrich conjecture for a more general class of ideals. We also obtain an upper bound on their Castelnuovo-Mumford regularity, extending Macaulay's bound and a theorem of Eisenbud-Huneke-Ulrich. We introduce higher syzygy bundles, which generalize the classical Green-Lazarsfeld syzygy bundle. The key ingredient is the relationship between the syzygies of partially linear ideals and the sheaf cohomology of higher syzygy bundles.