Cartwright–Sturmfels ideals associated to graphs and linear spaces

Abstract

Inspired by work of Cartwright and Sturmfels,in [15] we introduced two classes of multigraded ideals named after them. These ideals are defined in terms of properties of their multigraded generic initial ideals. The goal of this paper is showing that three families of ideals that have recently attracted the attention of researchers are Cartwright-Sturmfels ideals. More specifically, we prove that binomial edge ideals, multigraded homogenizations of linear spaces, and multiview ideals are Cartwright-Sturmfels ideals, hence recovering and extending recent results of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh [19], Ohtani [28], Ardila and Boocher [2], Aholt, Sturmfels, and Thomas [1], and Binglin Li [5]. We also propose a conjecture on the rigidity of local cohomology modules of Cartwright-Sturmfels ideals, that was inspired by a theorem of Brion. We provide some evidence for the conjecture by proving it in the monomial case.

Square-free Gröbner degenerations

“…Furthermore, a consequence of Theorem 1.3 is that, if in(I) is a square-free monomial ideal, then S/I satisfies Serre's condition (S r ) if and only if S/ in(I) does (see Corollary 2.11); the latter statement is false for generic initial ideals. As a further remark note that Conjectures 1.13 and 1.14 in [CDG18b] are indeed special cases of Theorem 1.3. The paper is structured as follows.…”

“…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.…”

When are multidegrees positive?

“…From a more algebraic point of view, multidegrees receive the name of mixed multiplicities (see Definition 2.7). More recent papers where the notion of multidegree (or mixed multiplicity) is studied are, e.g., [1,9,11,23,33,36,41,42,57].…”

Ideals of the Multiview Variety

“…Conca et al [5] and Li [15] also consider the vanishing ideal of the image of linear map from a projective space to a product of projective spaces. It is shown in [5] that this ideal is Cartwright-Sturmfels, meaning that its initial ideal is radical after a generic change of coordinates. Both of these works allow for projective spaces of arbitrary dimension.…”