Ranks of matrix factorizations and sheaf cohomology

TL;DR

This paper explores the relationship between matrix factorization ranks over hypersurface rings and sheaf cohomology on non-Fano projective hypersurfaces, proposing new conjectures linking algebraic and geometric properties.

Contribution

It introduces a graded version of a longstanding conjecture and connects it to a novel conjecture in algebraic geometry, bridging two mathematical areas.

Findings

Establishes a link between matrix factorization ranks and sheaf cohomology.

Proposes a new conjecture relating algebraic and geometric properties.

Suggests implications for the understanding of hypersurface structures.

Abstract

Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces.