Constraints for the normality of monomial subrings and birationality

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 $F\colon{\mathbb P}^{n-1}_k \dasharrow {\mathbb P}^{m-1}_k$ defined by $\mathbf{F}$ to be birational onto its image.