On the Weak Lefschetz Property for height four equigenerated complete intersections

TL;DR

This paper investigates the Weak Lefschetz Property for height four equigenerated complete intersections, proving it for degrees 3, 4, and 5, and exploring conditions under which it may fail, with implications for algebraic geometry.

Contribution

It provides new proofs for WLP in specific degrees, partial results in a broader range, and conjectures on failure cases, advancing understanding of complete intersections in algebraic geometry.

Findings

WLP holds for degrees 3, 4, and 5.

Partial maximal rank results in a wider range.

Conjecture on failure cases involving forms of the same degree.

Abstract

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree $d$ have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in $\mathbb P^3$, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for $d=3,4$ and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in~$\mathbb P^3$.