Toric ideals for high Veronese subrings of toric algebras. (English) Zbl 1273.13050

Shibuta, Takafumi

Ill. J. Math. 55, No. 3, 895-905 (2011).

Veronese subrings and Koszul algebras are very interesting topics. In a nice paper [Adv. Math. 109, No. 2, 168–187 (1994; Zbl 0839.13013)], D. Eisenbud, A. Reeves and 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.

Reviewer: Marcel Morales (Saint-Martin-d’Hères)

MSC:

13P10	Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

toric ideals; Veronese rings; toric algebras; Koszul algebras; Quadratic Gröbner basis

Citations:

Zbl 0839.13013

References:

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