Related papers: Exceptional sheaves on the Hirzebruch surface $\ma…

We find normal forms for del Pezzo surfaces of degree $2$ over algebraically closed fields of characteristic $2$. For each normal form, we describe the structure of the group of automorphisms of the surface. In particular, we classify all…

We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite…

We describe the effect of rational singularities on the Brauer group of a surface, and compute the Brauer groups of all singular del Pezzo surfaces over an algebraically closed field.

Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Koll\'ar. These surfaces have…

If $X$ is a singular del Pezzo surface of degree $d$ over a finite field $\mathbb{F}_{q}$ with only rational double point singularities, does there always exist a smooth $\mathbb{F}_{q}$-point on $X$? We show that this is true for $d\geq 3$…

I prove new local inequality for divisors on smooth surfaces, describe its applications, and compare it to a similar local inequality that is already known by experts.

Using Fedder's criterion, we classify all non-$F$-split del Pezzo surfaces of degree $1$. We give a necessary and sufficient criterion for the $F$-splitting of such del Pezzo surfaces in terms of their anti-canonical system.

Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We…

We consider a family of smooth del Pezzo surfaces of degree four and study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical.

We classify del Pezzo non-commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the…

In this paper we consider the classification of singularities (Du Val) and real structures (Wall) of weak Del Pezzo surfaces from an algorithmic point of view. It is well known that the singularities of weak Del Pezzo surfaces correspond to…

In this note we analyse the scrollar invariants of $k:1$ covers of $\mathbb P^1$ that factor through the normalisation of a nodal curve in the $m$-th Hirzebruch surface $\mathbb F_m$. We then give an existence theorem for nodal curves in…

We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field $\mathbb{F}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the…

We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose alpha-invariant of Tian is greater than 2/3.

The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence…

The paper consists of three parts. In the first of them different kinds stability are discussed. In particular, the stability concept with respect to nef divisor is introduced. A structure of rigid and superrigid vector bundles on smooth…

We prove several boundedness statements for geometrically integral normal del Pezzo surfaces $X$ over arbitrary fields. We give an explicit sharp bound on the irregularity if $X$ is canonical or regular. In particular, we show that wild…

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of…

We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo…

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.