Disconnected multigraded Hilbert schemes on ${\mathbb{P}}^2\times {\mathbb{P}}^1$

Abstract

We exhibit an infinite family of disconnected multigraded Hilbert schemes on the biprojective space ${\mathbb{P}}_{\mathbb{k}}^2{\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^1$. More precisely, for any integer $a\ge 2$ and $p_a(z_1,z_2)=2a{z}_1+a{z}_2+3a-2a^2\in \mathbb{Q}[z_1,z_2]$, the multigraded Hilbert scheme ${\mathrm{Hilb}}_{p_a}({\mathbb{P}}_{\mathbb{k}}^2{\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^1)$ is disconnected. As a consequence, there exist infinitely many disconnected Haiman-Sturmfels multigraded Hilbert schemes even for a standard bigrading on a polynomial ring in only five variables.