Related papers: Combinatorial Aspects of Classical Resolution of S…
We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset…
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a…
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…
In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties do not need to be complete. Fujino's completion theorem is the key to the main result. As applications,…
This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of a quasihomogeneous singularity. One of them is a new conjecture on the characteristic polynomial. It is an…
A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local…
In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
We consider 2+1-dimensional classical noncommutative scalar field theory. The general ansatz for a radially symmetric solution is obtained. Some exact solutions are presented. Their possible physical meaning is discussed. The case of the…
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…
This is a review article on the combinatorial aspects of the mixed Hodge structure of a Milnor fibre of the isolated hypersurface singularity. We give a purely combinatorial method to compute spectral pairs of the singularity under the…
We study contact resolutions of Jacobi structures which are contact on an open subset. We give several classes of examples, as well as classes for which it cannot exist.
We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club…
We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie…
We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the…
We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an…
We show that a version of the desingularization theorem of Hironaka holds for certain classes of infinitely differentiable functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the…
Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical…
In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new…
This paper is devoted to the resolution of singularities of holomorphic vector fields and of one-dimensional holomorphic foliations in dimension 3 and it has two main objectives. First, from the general perspective of one-dimensional…