Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces

TL;DR

This work studies the singularity category of Brieskorn-Pham hypersurfaces $R= k[X_1,\dots,X_4]/(\sum X_i^{p_i})$ through the stable category of arithmetically Cohen-Macaulay bundles on the associated Geigle-Lenzing space $\mathbb{X}$. The authors introduce $2$-extension bundles on $\mathbb{X}$, establish a correspondence with a distinguished class of Cohen-Macaulay $R$-modules, and construct a tilting object in $\underline{\mathsf{ACM}}\, \mathbb{X}$ whose endomorphism algebra is a $4$-fold tensor product of Nakayama algebras, yielding a concrete derived-equivalence description. They analyze the Picard group action on $2$-extension bundles and derive an explicit orbit-count formula that answers a higher version of a question of Kussin–Lenzing–Meltzer. The results provide a higher-dimensional extension of the KLM framework for weighted projective lines, enabling a tilting-theoretic and CM-representation-theoretic understanding of Brieskorn-Pham singularities via GL-spaces.

Abstract

We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating the stable category $\underline{\mathsf{ACM}} \, \mathbb{X}$ of arithmetically Cohen-Macaulay bundles on $\mathbb{X}$. We introduce the notion of $2$-extension bundles on $\mathbb{X}$, which is a higher dimensional analog of extension bundles on a weighted projective line of Geigle-Lenzing, and then establish a correspondence between $2$-extension bundles and a certain important class of Cohen-Macaulay $R$-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in $\underline{\mathsf{ACM}} \, \mathbb{X}$ consisting of $2$-extension bundles, whose endomorphism algebra is a $4$-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on $2$-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer.

Theorems & Definitions (93)

• Theorem 1.1: Theorem \ref{['correspondence of 2-Extension bundles']}, Corollary \ref{['bijections']}
• Theorem 1.2: Theorem \ref{['main theorem']}
• Theorem 1.3: Theorem \ref{['exact number']}
• Theorem 2.1: HIMO
• Theorem 2.2: HIMO
• Lemma 2.3: HIMO
• Definition 2.4
• Proposition 2.5
• proof
• Lemma 2.6