Monomial projections of Veronese varieties: new results and conjectures

In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong \mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variables where: (1) $\Omega_{n,d}$ contains all monomials supported in at most $s$ variables and, (2) $\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$. Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. For the family (1), we prove that $\mathbb{K}[\Omega_{n,d}]$ is quadratic when $s \ge \lceil \frac{n+2}{2} \rceil$. For the family (2), we completely characterize when $\mathbb{K}[\Omega_{2,d}]$ is quadratic in terms of the group $G \subset GL(3,\mathbb{K})$, and we prove that $\mathbb{K}[\Omega_{2,d}]$ is quadratic if and only if it is Koszul. We also provide large families of examples where $\mathbb{K}[\Omega_{n,d}]$ is G-quadratic.