Constraints for the normality of monomial subrings and birationality

Abstract:

Let $k$ be a field and let ${\mathbf F}\subset k[x_1,\ldots ,x_{n}]$ be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra $R[\mathbf {F}t]$ and the subring $k[\mathbf {F}]$. If the monomials in $\mathbb {F}$ have the same degree, one of the consequences is a criterion for the $k$-rational map $F\colon {\mathbb P}^{n-1}_k \rightdasharrow {\mathbb P}^{m-1}_k$ defined by $\mathbf {F}$ to be birational onto its image.