On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley

Abstract

In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any codimension $r$. In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form $h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $f(r)$, that $h_2$ may assume. In fact, we show that $$\underset{r\to \infty }{\lim }\frac{f(r)}{r^{2/3}}=6^{2/3}.$$ In general, we wonder whether our lower bound is sharp for all integers $e\ge 4$ and $r\ge 2$.

A few minor changes. To appear in Proc. of the AMS

References in corpus (1)

• Three questions in Gromov-Witten theory