Universal Gröbner Bases and Cartwright–Sturmfels Ideals

Abstract

We describe the universal Gröbner basis of the ideal of maximal minors and the ideal of 2-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.

Square-free Gröbner degenerations

“…Cartwright-Sturmfels ideals. Cartwright-Sturmfels ideals were introduced and studied by Conca, De Negri, Gorla in a series of recent papers[CDG15,CDG17,CDG18,CDG18b] inspired by the work[CS10]. We recall briefly their definition and main properties.…”

Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

“…Combining his result with the result of Giusti and Merle, one obtains a characterization of the graphs G such that I k m (X gen G ) is prime. (4) Generalizing the result of Boocher, in [9] and [10] it is proved that ideals of maximal minors of a matrix of linear forms that is either row or column multigraded is radical.…”

“…For this we need to introduce the concept of Sturmfels-Cartwright ideals. This concept was coined in [10] inspired by earlier work in [9] and [8]. It was further developed and applied to various classes of ideals in [11] and [12].…”

“…To prove that I is Cartwright-Sturmfels we have to prove that there is a Borel-fixed radical ideal J such that the K-polynomial of S/J is F (z). Taking into consideration the duality between Cartwright-Sturmfels ideals and Cartwright-Sturmfels * ideals discussed in [10], it is enough to exhibit a monomial ideal J whose generators are in the polynomial ring S = k[y 1 , y 2 , . .…”

Cartwright–Sturmfels ideals associated to graphs and linear spaces

“…As in the Z-graded setting, Z n -graded generic initial ideals are Borel fixed. We refer to [15,Section 1] for more details on multigraded generic initial ideals. We just recall that to any a ∈ n i=1 [d i ] 0 one can associate the Borel fixed prime ideal P a = (x ij : 1 ≤ i ≤ n and 1 ≤ j ≤ a i ) and that any Borel fixed prime ideal of S is of this form.…”