Cohen–Macaulay property and linearity of pinched Veronese rings

Abstract

In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules. The Veronese embedding injects the projective space Pn−1 into PN−1 by sending x = [x1 : x2 : · · · : xn] to the point with projective coordinates all possible monomials x1 1 . . . x in n of degree d, so N = ( n+d−1 d ) . The d-Veronese ring, S, is the coordinate ring of the image of the d-th Veronese embedding of Pn−1, with S = K[x1, . . . , xn]. The pinched Veronese map is another embedding of Pn−1, but this time the target space is PN−2 and the components of the image of x are all but one of the possible monomials. We denote such monomial by x. The coordinate ring of the latter image of Pn−1 is called pinched Veronese rings, Pn,d,m. The koszul property of the pinched Veronese rings was a trendy topic in literature. Peeva and Sturmfels asked whether the pinched Veronese ring P3,3,(1,1,1) is Koszul. A positive answer was given by Caviglia in [7], and then reproved by Caviglia and Conca in [8], and, after, in [9]; later, Tancer [21] generalized this result to Pn,n,(1,...,1). Vu used a combinatorial approach to prove that Pn,d,m is Koszul, unless d ≥ 3 and m is one of the permutations of (d− 2, 2, 0, . . . , 0), see [22]. 1991 AMS Mathematics subject classification. 13D02; 13D40; 05E99; 13C14, 13A02.