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 $\mathbb{P}^2_{\mathbb{k}} \times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}}$. More precisely, for any integer $a \ge 2$ and $p_a(z_1,z_2) = 2az_1+az_2+3a-2a^2 \in \mathbb{Q}[z_1,z_2]$, the multigraded Hilbert scheme ${\rm Hilb}_{p_a}(\mathbb{P}^2_{\mathbb{k}}\times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}})$ 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.

Disclosure

“2 YAIRON CID-RUIZ AI disclosure. OpenAI Codex, with GPT-5.5 and GPT-5.6, was used for exploratory computations that helped identify the counterexample for a = 2. The author conceived the theoretical framework, wrote the manuscript, and assumes full responsibility for its contents.”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:00:26.770159Z

Structural counts