卷中四次K3纤维Calabi-Yau三维簇的有限性定理
A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls
AI总结：
本文针对堆卷中以四次K3曲面为纤维的Calabi-Yau三维簇，应用Reid-Shepherd-Barron-Tai准则分类了n=1,2,3的情况，共得到14个形变族，补充了相关显式分类数据。
AI中文摘要：
我们研究Gross关于代数极小Calabi-Yau三维簇的有限性问题的受限形式：这类三维簇以四次K3曲面为纤维，且实现为堆卷$\boldsymbol{\text{PP}}^1\times[\boldsymbol{\text{PP}}^3/\boldsymbol{\text{Z}}_n]$中的反典范超曲面。除了文献\ucf1cite{MboyaSzendroi2023}中表1所列的10个直卷族外，我们引入了轨形卷$(\boldsymbol{\text{PP}}^1\times\boldsymbol{\text{PP}}^3)/\boldsymbol{\text{Z}}_n$，并应用Reid-Shepherd-Barron-Tai准则确定哪些卷存在典范反典范超曲面。对每个$n$仅出现有限个权向量；我们完全分类了$n=1,2,3$的情况并计算了所有Hodge数，总计给出14个形变族。Engel、Filipazzi、Greer、Mauri和Svaldi等人\ucf1cite{EngelFilipazziGreerMauriSvaldi2025}随后确立了纤维Calabi-Yau三维簇的有界性，解决了这类受限情形所例证的存在性问题；而我们在此提供的是剩余的显式分类：权向量、奇点类型及Hodge数据。
英文摘要：
We study a restricted form of Gross's finiteness problem for algebraic minimal Calabi--Yau threefolds: those fibred by quartic K3 surfaces and realised as anticanonical hypersurfaces in stacky scrolls $\PP^1\times[\PP^3/\ZZ_n]$. Beyond the ten straight-scroll families of \cite[Table~1]{MboyaSzendroi2023}, we introduce orbifold scrolls $(\PP^1\times\PP^3)/\ZZ_n$ and apply the Reid--Shepherd-Barron--Tai criterion to determine which admit canonical anticanonical hypersurfaces. Only finitely many weight vectors arise for each $n$; we classify $n=1,2,3$ completely and compute all Hodge numbers, giving fourteen deformation families in total. Engel, Filipazzi, Greer, Mauri and Svaldi \cite{EngelFilipazziGreerMauriSvaldi2025} have since established boundedness for fibred Calabi--Yau threefolds in general, settling the existence question this restricted case exemplifies; what remains, and what we supply here, is the explicit classification: weight vectors, singularity types, and Hodge data.
