Demailly's Conjecture and the Containment Problem

Abstract

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.

Symbolic powers of generalized star configurations of hypersurfaces

“…Proof. The Harbourne-Huneke type containment follows from (9), since both d and µ h are positive integers, while k can be specially taken to be h. By applying the identical arguments in the proof of [4,Corollary 3.7], we see that the Demailly-like bound also holds.…”

“…In this work, we also study the Harbourne-Huneke containment problem, which was originally raised in [14]. After [4], a generalized version can be stated as: given a homogeneous ideal I of big height h in a standard graded ring with the maximal homogeneous ideal m, does the inequality…”

“…Furthermore, we have seen in the proof of [17,Proposition 4.7] that c 0 = h precisely when b = 1 or a = bs. The b = 1 case is just the standard star configuration case studied in [4]. Meanwhile, the a = bs case is the trivial one when the ideal is actually principal.…”

Rees algebras of ideals of star configurations

“…From a commutative algebra perspective, ideals defining star configurations represent an interesting class, since a great amount of information is known about their free resolutions, Hilbert functions and symbolic powers (see for instance [13,14,11,22,24,2,3,28,23]). In this article we study their Rees algebras, about which little is currently known (see for instance [18,12,26,5]).…”