Linear syzygies and birational combinatorics

Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = k[x1, . . . , xn] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension k[F ] ⊂ R, where R is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theoretic birationality criterion in characteristic zero obtained in [1].