Related papers: Normal del Pezzo surfaces containing a nonrational…

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic…

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y…

We prove that for Du Val del Pezzo surfaces of degree one with Picard rank two, the existence of an anticanonical polar cylinder implies the ample polar cylindricity.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a normal projective surface over $k$ with canonical singularities whose anti-canonical divisor is nef and big. We prove that $X$ is globally $F$-regular except for…

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.

We classify del Pezzo surfaces with Picard number is equal to one and with four log terminal singular points.

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $\varphi$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in…

We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.

We study del Pezzo fibrations of degree 1 with terminal singularities. A connection between singularities on del Pezzo surfaces of degree 1 and Kodaira's classification of elliptic singular fibers will be studied in this paper. By this…

Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as…

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if…

Inspired by the recent progress by Coates-Corti-Kasprzyk et al. on Mirror Symmetry for del Pezzo surfaces, we show that for any positive integer k the deformation families of del Pezzo surfaces with a single 1/k(1,1) singularity (and no…

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…

Let $\bar{Y}$ be a normal surface that is the canonical $\mu_2$- or $\alpha_2$-covering of a classical or supersingular Enriques surface in characteristic $2$. We determine all possible configurations of singularities on $\bar{Y}$, and for…

This article wants to show two things: first, that certain problems in Diophantus' Arithmetica lead to equations defining del Pezzo surfaces or other rational surfaces, while certain others lead to K3 surfaces; second, that Diophantus' own…

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.

In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain…

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.

For each integer d=2,3,4, there exists a field F with cohomological dimension 1 and a del Pezzo surface of degree d over F having no rational point. Proofs use the theorem of Merkur'ev and Suslin, the Riemann-Roch theorem on a surface and…