Nilpotent operators and weighted projective lines. (Q2872161)

The article under review approaches the classification problem for invariant subspaces of nilpotent linear operators (studied by C. M. Ringel and M. Schmidmeier) through categorical methods, obtaining a wide range of results based on the study of vector bundles on a weighted projective line.NEWLINENEWLINE Let \(\mathbb X=\mathbb X(2,3,p)\), \(p\geq 2\), be the weighted projective line of the corresponding weights and denote the category coh-\(\mathbb X\) of coherent sheaves on \(\mathbb X\), obtained by applying Serre's construction to the triangle singularity \(x_1^2+x_2^3+x_3^p\). The Picard group of \(\mathbb X\) is the rank one Abelian group \(\mathbb L=\mathbb L(2,3,p)\) on three generators and the corresponding relations and, up to isomorphism, the line bundles are given by the system of twisted structure sheaves \(\mathcal Ox\), \(x\in\mathbb L\), which is denoted by \(\mathcal L\). Then, the stable triangulated category \(\underline{\text{vect}}\text{-}\mathbb X/[\mathcal L]\) has a tilting object \(T\) which is constructed in the appendix of the article, and has Serre duality.NEWLINENEWLINE The goal of the article is to show that the functor NEWLINE\[NEWLINE\Phi\colon\mathrm{vect-}\mathbb X\to\mathrm{mod-}\underline{\mathcal P},\quad E\mapsto\underline{\mathcal P}(-,E)NEWLINE\]NEWLINE induces an equivalence of the bounded derived categories \(D^b(\mathrm{coh-}\mathbb X)\) and \(D^b(\mathrm{mod-}A)\), where \(\mathcal P\) is the class of persistent line bundles in \(\mathcal L\) and \(A\) is the finite dimensional endomorphism algebra of \(T\), which is the representation-finite Nakayama algebra \(A(2(p-1),3)\) of certain quiver with nilpotency relations, which completes a relating picture between triangle singularities, the invariant subspace problem and representation theory of quivers.NEWLINENEWLINE The result allows to prove properties of the corresponding categories such as the action of the Picard group, the construction of tilting objects, the calculation of the fractional Calabi-Yau dimension, the Euler characteristic and finally to show that the categories form an ADE-chain, for \(p\geq 2\). A special role plays the case \(p=6\), having Euler characteristic zero, and whose classification of indecomposable bundles over the weighted projective line is very similar to the classical one by Atiyah of bundles over elliptic curves.