Poincaré series and Coxeter functors for Fuchsian singularities

We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincaré series. This uses Coxeter elements involving Eichler-Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories.     

Deformations of canonical singularities

We prove that small deformations of canonical singularities are canonical.

     

Symmetries and apparent singularities for the simplest Fuchsian equations

We consider the simplest Fuchsian second-order equations with particular attention to the role of apparent singularities. We show the relation to the Painlevé equation and follow the matrix formulation of the problem.     

Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Semi-invariants of canonical algebras

We describe the algebras of semi-invariants on the varieties of regular representations of canonical algebras. In particular, we show that these algebras are polynomial algebras or complete intersections. Received: 29 March 1999     

On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities

We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables $\stackrel{?}{X}(t,z)$ for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of $\mathbb{C}$, is $\stackrel{?}{X}(t,z)$. The small divisors? effect is an increase in the order of q-exponential growth and the appearance of a power of the factorial in the corresponding q-Gevrey bounds in the asymptotics.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

The canonical sheaf of Du Bois singularities

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π_∗ωX^′(G)?ωX for a log resolution π:X^′→X, where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.     

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.     

Moduli algebras of some non-semiquasihomogeneous singularities

Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.     

Indecomposable modules for domestic canonical algebras

We show that-up to precisely one-each exceptional module over a domestic canonical algebra of quiver type over a field k can be represented by matrices whose entries are just 0 and 1. In the case we calculate the matrices of these representations explicitly.     

Extended dyer-lashof algebras and modular coinvariants

Given an increasing sequence of integersN=(0,n ₁,n ₂,...), a functorG _N is constructed from the category of based spaces of the homotopy type ofCW compleces and based maps to a subcategory of in analogy to May's approximation modelC. A family of homology operationsRN is associated toG _N and its algebraic structure is described in terms of modular coinvariants of parabolic subgroups.     

Extended Rees algebras and mixed multiplicities

Partially supported by the general research fund at the University of Kansas     

Derived equivalences of selfinjective algebras preserve singularities

We prove that the derived equivalences (more generally the stable equivalences of Morita type) of finite dimensional selfinjective algebras over algebraically closed fields preserve the types of singularities in the orbit closures of module varieties. As an application, we obtain that the orbit closures in the module varieties of the Brauer tree algebras are normal and Cohen-Macaulay. Mathematics Subject Classification (2000):14B05, 14L30, 16D50, 16G20