A non-unimodal codimension 3 level $h$-vector

TL;DR

This paper constructs explicit examples of codimension 3 level $h$-vectors that are non-unimodal and have multiple maxima, demonstrating the existence of level algebras without the Weak Lefschetz Property, thus answering longstanding open questions.

Contribution

It provides the first known examples of non-unimodal codimension 3 level $h$-vectors and constructs level algebras with any number of maxima, also showing some lack the WLP.

Findings

Explicit non-unimodal level $h$-vector example $(1,3,6,10,15,21,28,27,27,28)$

Construction of level algebras with exactly $N$ maxima for any positive integer $N$

Existence of codimension 3 level algebras of type 3 without the Weak Lefschetz Property

Abstract

$(1,3,6,10,15,21,28,27,27,28)$ is a level $h$-vector! This example answers negatively the open question as to whether all codimension 3 level $h$-vectors are unimodal. Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose $h$-vectors have exactly $N$ ` ` maxima", for any positive integer $N$. These non-unimodal $h$-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before. In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP.