Interpolation and the weak Lefschetz property

Abstract

Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo-Mumford regularity of the corresponding ideals. As applications we establish new instances of conjectures by Chudnovsky and by Demailly on the Waldschmidt constant. Moreover, we show that conjectures on the failure of the weak Lefschetz property by Harbourne, Schenck, and Seceleanu as well as by Migliore, Miró-Roig, and the first author are true asymptotically. The latter also relies on a new result about Eulerian numbers.

Squeezed complexes

“…There is a rich literature on the problem of deciding whether a Cohen-Macaulay algebra has one of the Lefschetz properties. It has been studied from many different points of view, applying tools from representation theory, topology, vector bundle theory, lozenge tilings, splines, hyperplane arrangements, inverse systems, differential geometry, among others (see, e.g., [4,5,6,7,9,12,13,14,20,21,26,27]).…”

On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

“…By Proposition 3.2, we have D = 0 for d = 15e + 2r, e and r are non-negative integers such that 2 ≤ r ≤ 8. Using (4.1), we can also show that D = 0 for d ≥ 4 and d = 5, 7,9,11,13,18,20. We now need to prove D = 0 for d = 5, 7,9,11,13,18,20.…”

“…Using (4.1), we can also show that D = 0 for d ≥ 4 and d = 5, 7,9,11,13,18,20. We now need to prove D = 0 for d = 5, 7,9,11,13,18,20. With the notations as in the case 2, one has…”

Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points

“…After that, a similar approach using specialization as in [BGHN20], see also [BGHN22], yields the results for a small number of general points. Note that the Waldschmidt constant for defining ideals of up to N + 3 generic points are computed in [DHSTG14] and Harbourne-Huneke Containment as well as Chudnovsky's Conjecture would follow easily, see also [NT19]. Hence, in this manuscript, we are interested in ideals defining at least N + 4 many generic points when N ≥ 4.…”

“…Some well known values of Waldschmidt constants of defining ideals of small number of points, see also [NT19].…”