Miller–Villarreal conjectures on generators of Gorenstein ideals

Let $R$ be a polynomial ring over a field, and let $I$ be a homogeneous Gorenstein ideal of codimension $g\geq 3$ and initial degree $p\geq 2$. Write $I_p$ for the degree-$p$ component of $I$, let $\nu(I_p)$ denote the number of minimal generators of $I$ in degree $p$, and set

$$\nu_0={p+g-1\choose g-1}-{p+g-3\choose g-1}.$$

Also write $e(R/I)$ for the multiplicity of $R/I$, and let $e(g,p)$ denote the multiplicity of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$.

Miller–Villarreal conjectures. One always has

$$\nu(I_p)\leq\nu_0,$$

and only certain values of $\nu(I_p)$ are possible. If $\nu(I_p)=\nu_0$, then $I$ is extremal in the sense of the cited reference, equivalently

$$e(R/I)=e(g,p)={g+p-1\choose g}+{g+p-2\choose g}.$$

Consequently, $\nu(I_p)=\nu_0$ implies $I=(I_p)$.

The paper studies restrictions on the minimal generators of homogeneous Gorenstein ideals. The supplied text labels these assertions as conjectures; no resolution status is given here.