Related papers: Elliptic singularities and threefold flops in posi…

We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to…

We investigate fibrations by non-hyperelliptic curves of arithmetic genus three and geometric genus one in characteristic two. Assuming that there is only one moving singularity and that its image in the Frobenius pullback of the fibration…

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set…

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it…

We point out an interesting relation between hypersurface elliptic singularities and log Enriques surfaces: with a few exceptions, every hypersurface elliptic singularity define some klt log Enriques surface $(S,Diff)$. In many cases, the…

Let X be a projective, equidimensional, singular scheme over an algebraically closed field. Then the existence of a geometric smoothing (i.e. a family of deformations of X over a smooth base curve whose generic fibre is smooth) implies the…

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a…

In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples…

Algebraically simply connected surfaces of general type with p_g=q=0 and 1\le K^2\le 4 in positive characteristic (with one exception in K^2=4) are presented by using a Q-Gorenstein smoothing of two-dimensional toric singularities, a…

Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the…

We define and calculate the weighted multiplicities of non Gorenstein terminal singularities on threefolds and some quotient singularities. We improve freeness conditions on threefolds.

In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve…

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical…

In this paper we study the possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt {-3})$.

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface $X\subset \bbP^{4}$ of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type…

We study codimension one smooth foliations with Morse type singularities on closed ma-nifolds. We obtain a description of the manifold in case the number of centers in greater then the number of saddles. This result relies on and extends…

We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.

We survey some of the recent works on the geometry of del Pezzo surfaces over imperfect fields, with applications to 3-dimensional del Pezzo fibrations in positive characteristic. We place particular emphasis on cases where the general…

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured…