Koszul homology and syzygies of Veronese subalgebras

Abstract

A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.

17 pages, minor modifications

References in corpus (4)

• Syzygies, multigraded regularity and toric varieties
• Free resolutions over commutative Koszul algebras
• The cone of Betti diagrams of bigraded artinian modules of codimension two
• Some effects of Veronese map on syzygies of projective varieties

Cited by in corpus (16)

• Asymptotic syzygies of algebraic varieties
• On the index of powers of edge ideals
• A quick proof of nonvanishing for asymptotic syzygies
• Representation stability for syzygies of line bundles on Segre--Veronese varieties
• Retracts and algebraic properties of cut algebras
• On the n-th row of the graded Betti table of an n-dimensional toric variety
• Free resolutions over commutative Koszul algebras
• Subadditivity of syzygies of Koszul algebras
• On Syzygies, degree, and geometric properties of projective schemes with property ${\text{N}}_{3,p}$
• On the Betti numbers of edge ideal of skew Ferrers graphs
• Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties
• Arithmetic properties of the first secant variety to a projective variety
• Hilbert functions of Cox rings of del Pezzo surfaces
• Castelnuovo-Mumford regularity of Koszul cycles and Koszul homologies
• Regularity bounds for Koszul cycles
• $N_6$ property for third Veronese embeddings