Monomial ideals and the failure of the Strong Lefschetz property

We give a sharp lower bound for the Hilbert function in degree d of artinian quotients $$\Bbbk [x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree $$d \ge 2$$ d ≥ 2 . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

Representation theory of symmetric groups and the strong Lefschetz property

We investigate the structure and properties of an Artinian monomial complete intersection quotient [Formula: see text]. We construct explicit homogeneous bases of [Formula: see text] that are compatible with the [Formula: see text]-module structure for [Formula: see text], all exponents [Formula: see text] and all homogeneous degrees [Formula: see text]. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the [Formula: see text]-module decomposition of homogeneous subspaces.

Line arrangements and configurations of points with an unexpected geometric property

We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union$X$of fat points imposes on the complete linear system of curves in$\mathbb{P}^{2}$of fixed degree$d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by$X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.