Exceptional Unimodal Singularities Research Articles

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations.This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

On the deformation of the exceptional unimodal singularities

On the deformation of the exceptional unimodal singularities

A note on distinguished bases of singularities

For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter–Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter–Dynkin diagrams like the exceptional unimodal singularities.

Orbifold Jacobian Algebras for Exceptional Unimodal Singularities

This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund–Hubsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.

On a certain generalization of triangle singularities

Triangle singularities are Fuchsian singularities associated with von Dyck groups, which are index two subgroups of Schwarz triangle groups. Hypersurface triangle singularities are classified by Dolgachev, and give 14 exceptional unimodal singularities classified by Arnold. We introduce a generalization of triangle singularities to higher dimensions, show that there are only finitely many hypersurface singularities of this type in each dimension, and give a complete list in dimension 3.

A geometric definition of Gabrielov numbers

Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $$G$$ of $$\mathrm{SL}(3,{\mathbb {C}})$$ using the Gabrielov numbers of the cusp singularity and data of the group $$G$$ . Here we consider a crepant resolution $$Y \rightarrow {\mathbb {C}}^3/G$$ and the preimage $$Z$$ of the image of the Milnor fibre of the cusp singularity under the natural projection $${\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G$$ . Using the McKay correspondence, we compute the homology of the pair $$(Y,Z)$$ . We construct a basis of the relative homology group $$H_3(Y,Z;{\mathbb {Q}})$$ with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.

A Note on Exceptional Unimodal Singularities and K3 Surfaces

This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).

On Arnold’s 14 ‘exceptional’ $ \mathcal{N} = 2 $ superconformal gauge theories

We study the four-dimensional superconformal $ \mathcal{N} = 2 $ gauge theories engineered by the Type IIB superstring on Arnold’s 14 exceptional unimodal singularities (a.k.a. Arnold’s strange duality list), thus extending the methods of arXiv:1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several ‘strongly coupled’ chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with ĉ < 2 (more generally, for each $ \mathcal{N} = 2 $ superconformal theory with a finite BPS chamber whose chiral primaries have dimensions of the form $ {{\mathbb{N}} \left/ {\ell } \right.} $ ).

Strange duality of weighted homogeneous polynomials

AbstractWe consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold’s strange duality between the 14 exceptional unimodal singularities.

Extended canonical algebras and Fuchsian singularities

The authors introduce a new class of finite dimensional algebras called extended canonical, and determine the shape of their derived categories. Extended canonical algebras arise from a canonical algebra Λ by onepoint extension or coextension by an indecomposable projective module. Our main results concern the case of negative Euler characteristic of the corresponding weighted projective line $${\mathbb{X}}$$ ; more specifically we establish, for a base field of arbitrary characteristic, a link to the Fuchsian singularity R of $${\mathbb{X}}$$ which for the base field of complex numbers is isomorphic to an algebra of automorphic forms. By means of a recent result of Orlov we show that the triangulated category of the graded singularities of R (in the sense of Buchweitz and Orlov) admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra. Of particular interest are those cases where the attached Coxeter transformation has spectral radius one. A K-theoretic analysis then shows that this happens exactly for 38 cases including Arnold’s 14 exceptional unimodal singularities. The paper is related to recent independent work by Kajiura, Saito and Takahashi.

Duality of Weights, Mirror Symmetry and Arnold's Strange Duality

A duality of weight systems which corresponds to Batyrev's toric mirror symmetry is given. It is shown that Arnold's strange duality for exceptional unimodal singularities reduces to this duality.