Toric ideals for high Veronese subrings of toric algebras (Q351813)

Veronese subrings and Koszul algebras are very interesting topics. In a nice paper [Adv. Math. 109, No. 2, 168--187 (1994; Zbl 0839.13013)], \textit{D. Eisenbud, A. Reeves} and \textit{B. Totaro} proved that if \(S\) is a graded polynomial ring over a field \(K\), and \(I\) a homogeneous ideal, then up to generic coordinates the initial ideal of \(n\)-Veronese transform of the ideal \(I\), \(\text{in}(V_n(I)) \) is generated by quadratic monomials for \(n \) large enough. An effective bound was given in terms of the Castelnuovo-Mumford regularity of \(I\). In the paper under review, the author considers a toric standard graded ideal \(I\) in a polynomial ring, and proves that for some monomial order \(\prec \), he can define a monomial order such that \(\text{in}(V_n(I)) \) is generated by quadratic monomials for \(n \) large enough, a bound is given in terms of \(\text{in}_{\prec }(I), \) this result seems to be effective and follows from a more general result.