Related papers: Note on generic singularities of planar flat 3-web…

We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.

Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…

We investigate the problem of existence of degenerations of surfaces in $\mathbb P^3$ with ordinary singularities into plane arrangements in general position.

By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…

We provide a complete classification of all algebras of generalised dihedral type, which are natural generalizations of algebras which occurred in the study of blocks with dihedral defect groups. This gives a description by quivers and…

For any k<2n we construct a complete system of invariants in the problem of classifying singularities of immersed k-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

Let S be a complex smooth projective surface and L be a line bundle on S. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system |L| with prescribed singularities is…

We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable…

We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the…

We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong…

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper…

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with…

Isolated hypersurface singularities come equipped with distinguished bases of their Milnor lattices and with upper triangular integral matrices, which are called here distinguished matrices. These matrices form an orbit of a braid group and…

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean…

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…

In this paper we study plus-one generated arrangements of conics and lines in the complex projective plane with simple singularities. We provide several degree-wise classification results that allow us to construct explicit examples of such…

We describe those Weil divisors of cyclic quotient surface singularities which are (abstract) $r$--tuple curve singularities.

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are…

We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular…