2-hereditary algebras and almost Fano weighted surfaces

2016/04/20 by Daniel Chan, Chan, Daniel · 2 citations
Mathematics · #14J45 #16G10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1604.06141

openalex publication_date 2016/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Tilting bundles T on a weighted projective line \mathbbX have been intensively studied by representation theorists since they give rise to a derived equivalence between \mathbbX and the finite dimensional algebra End T. A classical result states that if End T is hereditary, then \mathbbX is Fano and conversely, for every Fano weighted projective line, there exists a tilting bundle T with End T hereditary. In this paper, we examine the question of when a weighted projective surface has a tilting bundle whose endomorphism ring is 2-hereditary in the sense of Herschend-Iyama-Oppermann. It is natural to conjecture that they are the almost Fano weighted surfaces, weighted only on rational curves, and we give evidence to support this.

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• Weak del Pezzo surfaces yield 2-hereditary algebras and 3-Calabi-Yau algebras
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