Gotzmann monomial ideals

Abstract

A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let $R=K[x_1,\dots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $M^d$ the set of monomials of $R$ of degree $d$. A subset $V\subset M^d$ is said to be a Gotzmann subset if the ideal generated by $V$ is a Gotzmann monomial ideal. In the present paper, we find all integers $a > 0$ such that every Gotzmann subset $V\subset M^d$ with $|V|=a$ is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of $K[x_1,x_2,x_3]$.