Related papers: Which rational double points occur on del Pezzo su…

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are…

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined…

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the…

In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…

We study the distribution of the Brauer group and the frequency of the Brauer--Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations…

We obtain a formula for the number of genus two curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done by extending the…

Suppose V is a surface over a number field k that admits two elliptic fibrations. We show that for each integer d there exists an explicitly computable closed subset Z of V, not equal to V, such that for each field extension K of k of…

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type…

Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever…

Detailed illustration of the method for calculating the Chow group of a rational surface over a local field [math.AG/0302157 (th.~4)], applied to a certain del Pezzo surface of degree~4. Involves the construction of a regular integral model…

This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois…

The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the…

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length…

In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\text{Gal} ( K^{\text{sep}}/K)$ on the graph of…

We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii)…

We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove…

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $\mathbb{S}_r$ arising by blowing-up of $\mathbb{P}^2$ at $r$ general points, with $ r \in \{1, \dots, 8 \}$. Basic questions when…

We propose the new construction of complex surfaces with $h^{1,0} = h^{2,0} = 0$ from smoothings of normal crossing surfaces with non-collapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the…

We settle a question that originates from results and remarks by Koll\'ar on extremal ray in the minimal model program: In positive characteristics, there are no Mori fibrations on threefolds with only terminal singularities whose generic…