Related papers: Smoothable del Pezzo surfaces with quotient singul…

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a…

Let S_r be the blow-up of P^2 in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. For r <= 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S_8) is calculated up to…

Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an…

We prove new local inequality for divisors on surfaces and utilize it to compute $\alpha$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$,…

We show that smooth quintic del Pezzo threefolds over arbitrary base schemes are classified by non-degenerate ternary symmetric bilinear forms. Then we describe the automorphism group schemes, the Hilbert schemes of lines and the orbit…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by…

I construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity q>0. Such things can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo…

For curves singularities the dimension of smoothing components in the deformation space is an invariant of the singularity, but in general the deformation space has components of different dimensions. We are interested in the question what…

We introduce a class of noncommutative surfaces called Artin--Schelter surfaces of del Pezzo types, which contains del Pezzo surfaces as special cases. We show that the moduli stacks of marked Artin--Schelter surfaces of del Pezzo types are…

We extend the results of Pareschi on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree 1, where we explicitly…

In this article we present a unified way to smooth certain multiple structures called ropes on smooth varieties. We prove that most ropes of arbitrary multiplicity, supported on smooth curves can be smoothed. By a rope being smoothable we…

We classify codimension 2 well-formed and quasi-smooth weighted complete intersection del Pezzo surfaces.

We classify all normal stable Horikawa surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities. Furthermore, we provide a criterion for their global $\mathbb{Q}$-Gorenstein smoothability and describe the boundary…

The $k$-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface $X$ corresponds to a regular unimodular triangulation $D$ of the polytope defining $X$. If the secant ideal of the…

We give an algorithm that, for a given value of the geometric genus $p_g,$ computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base…

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo…

Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $exp(\frac{2\pi i}{k})$ is a zero of the Alexander polynomial. Then we show that the…

Del Pezzo surfaces over C with log terminal singularities of index \le 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of…