Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type

By A. Takahashi. 2007

47 citations recorded by the source corpus.

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. 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 the joint paper with Kajiura and Saito. 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 can be applied.

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Citations in the corpus

• Strange duality of weighted homogeneous polynomials (2010, background): “ry which gives the systematic construction of mirror pairs of Calabi–Yau manifolds. Therefore, we may expect that the topological mirror symmetry can also be categorified to the following: Conjecture ([T1][T2]). Let f(x,y,z) be an invertible polynomial (see Section 1 for the definition). (i) There sho”
• Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “evich [7] for Calabi-Yau manifolds, and later generalized to more general classes of manifolds [8, 5, 14]. Homological mirror symmetry for singularities are studied by Takahashi and his collaborators [19, 3, 4, 18]. Okada [9] also discusses homological mirror symmetry for Brieskorn-Pham singularitie”