Linear syzygies and birational combinatorics

Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = kx ₁,…, x _n ] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension kF] ? R ^(d), where R ^(d) is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygytheoretic birationality criterion in characteristic zero obtained in 1].