AI 中文总结
本研究针对双射影空间P²×P¹上的多重分次Hilbert概型连通性问题，通过构造特定多项式对应的概型族，证明存在无穷多个不连通的该类概型，拓展了低变量多项式环下的相关结论。
AI 中文摘要
我们构造了双射影空间$\u2119^2_\u1d52 \times_\u1d52 \u2119^1_\u1d52$上一族无穷多个不连通的多重分次Hilbert概型。更确切地说，对任意整数$a \ge 2$以及多项式$p_a(z_1,z_2) = 2az_1+az_2+3a-2a^2 \in \u211a[z_1,z_2]$，多重分次Hilbert概型${\rm Hilb}_{p_a}(\u2119^2_\u1d52\times_\u1d52 \u2119^1_\u1d52)$是不连通的。由此可得，即使对于仅含五个变量的多项式环上的标准双分次，也存在无穷多个不连通的Haiman-Sturmfels多重分次Hilbert概型。
英文摘要
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.
