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VOL. 59 | 2010 Weighted projective lines associated to regular systems of weights of dual type
Atsushi Takahashi

Editor(s) Masa-Hiko Saito, Shinobu Hosono, Kōta Yoshioka

Abstract

We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights.

The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries.

The main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in [5]. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne–Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle–Lenzing [2] can be applied.

Information

Published: 1 January 2010
First available in Project Euclid: 24 November 2018

zbMATH: 1213.14075
MathSciNet: MR2683215

Digital Object Identifier: 10.2969/aspm/05910371

Subjects:
Primary: 14J33
Secondary: 32S25, 53D37

Rights: Copyright © 2010 Mathematical Society of Japan

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