Monomial projections of Veronese varieties: New results and conjectures

Abstract

In this paper, we consider the homogeneous coordinate rings A(Yn,d)≅K[Ωn,d] of monomial projections Yn,d of Veronese varieties parameterized by subsets Ωn,d of monomials of degree d in n+1 variables where: (1) Ωn,d contains all monomials supported in at most s variables and, (2) Ωn,d is a set of monomial invariants of a finite diagonal abelian group G⊂GL(n+1,K) of order d. Our goal is to study when K[Ωn,d] is a quadratic algebra and, if so, when K[Ωn,d] is Koszul or G-quadratic. For the family (1), we prove that K[Ωn,d] is quadratic when s≥⌈n+22⌉. For the family (2), we completely characterize when K[Ω2,d] is quadratic in terms of the group G⊂GL(3,K), and we prove that K[Ω2,d] is quadratic if and only if it is Koszul. We also provide large families of examples where K[Ωn,d] is G-quadratic.