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 weig...

Given a smooth (as stack) Calabi-Yau hypersurface X in a weighted projective space, we consider the functor G which is the composition of the following two autoequivalences of D(X): the first is induced by the spherical object OX , while the second is tensoring by OX(1). The main result of the paper is that the composition of G with itself w times, where w is the sum of the weights, is isomorph...

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as sp...

We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component X of Omin ∩ n+ (where Omin is the (Zariski) closure of the minimal nilpotent orbit of sp2n and n+ is the Borel subalgebra of sp2n) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of U(sp2n), which contains the maxi...

We construct the first examples of families of bad Riemannian orbifolds which are isospectral with respect to the Laplacian but not isometric. In our case these are particular fixed weighted projective spaces equipped with isospectral metrics obtained by a generalization of Schüth’s version of the torus method.

In this article, we prove the following results. • We show a mirror theorem : the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to a specific Laurent polynomial, • We show a reconstruction theorem, that is, we can reconstruct in an algorithmic way the full genus 0 Gromov-Witten potential from the 3-point invariants.

In [1] some quotients of one-parameter families of Calabi-Yau varieties are related to the family of Mirror Quintics by using a construction due to Shioda. In this paper, we generalize this construction to a wider class of varieties. More specifically, let A be an invertible matrix with non-negative integer entries. We introduce varieties XA and MA in weighted projective space and in P, respect...

We compute the Hodge numbers of the polarised (pure) variation of Hodge structure V = grn−1 R f! Z of the Landau-Ginzburg model f : Y → C mirror-dual to a weighted projective space wPn in terms of a variant of Reid’s age function of the anticanonical cone over wPn. This implies, for instance, that wPn has canonical singularities if and only if hn−1,0 V = 1. We state a conjectural formula for th...

Let X be a weighted projective line and CX the associated cluster category. It is known that can realized as generalized category of quiver with potential. In this note, under assumption has at most three weights or tubular type, we prove if C(Q,W) Jacobi-finite non-degenerate potential (Q,W) shares 2-CY tilted algebra CX, then triangle equivalent to CX. As byproduct, determined by its provided...