A note on generators of least degree in Gorenstein ideals

Abstract:

Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge 3$ and initial degree $p\ge 2$. We prove that the number of minimal generators $\nu (I_p)$ of $I$ that are of degree $p$ is bounded above by $\nu _0=\binom {p+g-1}{g-1}-\binom {p+g-3}{g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $\nu (I_p)=\nu _0$.