Containment results for ideals of various configurations of points in

Abstract

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in P N . In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.

A counterexample to the containment I(3) ⊂ I2 over the reals

“…In algebraic geometry and in commutative algebra there has been recently a lot of interest in comparing usual (algebraic) and symbolic powers of homogeneous ideals, see for example [2], [10], [3]. If I ⊂ K[x 0 , x 1 , .…”

“…If the number of triple points is high when related to the number of lines, then F will be the only element of degree s in I (3) . On the other hand, the generators of I should be of relatively high degree.…”

A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in P2

“…More precisely, it is proven therein that removing any single point from among the 13 points of the finite projective plane P 2 F 3 together with all the lines passing through the removed point yields on the remaining 12 points the same incidence structure exhibited by the dual Hesse configuration of [8]. It is worth noting, however, that although they share the same combinatorial data and the property that I (3) I 2 , the dual Hesse configuration of [8] and the characteristic 3 counterexample of [2] behave very differently when viewed from the perspective of algebraic-geometric invariants associated to them (see [7] for a detailed account of the differences and computations of the resurgence for these counterexamples).…”

“…The entire incidence structure of the 12 points and 9 lines is projectively dual to the classical Hesse configuration given by the 9 flexes of a general plane cubic together with the 12 lines passing through pairs of these flexes. A characteristic 3 analogue of the counterexample of [8] was later given in [2]. More precisely, it is proven therein that removing any single point from among the 13 points of the finite projective plane P 2 F 3 together with all the lines passing through the removed point yields on the remaining 12 points the same incidence structure exhibited by the dual Hesse configuration of [8].…”

On the containment hierarchy for simplicial ideals

“…[2], [3], [10]. The first counterexample to the I (3) ⊂ I 2 containment for an ideal of points in P 2 announced in [6] has prompted another series of papers [1], [11], [13]. In all these works the authors study containment relations of the type I (m) ⊂ I r for a fixed homogeneous ideal I.…”