Related papers: Singularities in positive characteristic, stratifi…
We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the…
We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction…
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $\phi \colon X \to \mathbb{P}^n$…
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are…
We obtain sufficient conditions for the vanishing of higher homotopy groups of the complements to hypersurfaces in ${\mathbb C}^n$ in terms of the behavior at infinity and relate the monodromy of non isolated singularities to the position…
We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of…
We consider the locus of $r$-tuples of homogeneous forms of some fixed degree whose common vanishing locus in $\mathbb{P}^r$ is positive dimensional. We show that any component of maximal dimension of that locus either consists of…
The concept of differential uniformity was recently extended to the $c$-differential uniformity. An interesting problem in this area is the construction of functions with low $c$-differential uniformity and a lot of research has been done…
A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant…
We construct an iterative method for factorising small strict n-categories into a unique (up to isomorphism) collection of small 1- categories. Following this we develop the theory to include a large class of $\infty$-categories. We use…
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…
The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems…
We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic…
Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, endowed with an ample line bundle L. We introduce a general notion of (possibly singular) semipositive (or…
This paper generalizes existing methods to derive stronger bounds on the modality of hypersurface singularities. Our results demonstrate that each sudden jump in the extended Tjurina number necessarily increases the modality. Furthermore,…
We study the topology of some simple infinite dimensional singularities arising from spaces of \emph{algebraic formal loops}. We prove that in some simple cases the natural analogue of nearby cycles cohomology for a function on the loop…
We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$.…
In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect…
The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…