Related papers: Full Degree Two del Pezzo Surfaces over Small Fini…
In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.
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.
Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$),, and asked whether such surfaces always have a closed point of degree…
We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and…
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 discuss the rational points on del Pezzo surface of degree 1 and 2 over any finite field $\mathbb F_q$, and give out the explicit equations of del Pezzo surfaces that have unique rational point.
We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a…
We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and V\'arilly-Alvarado, who had proved this for all but three surfaces. Over general…
We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and V\'arilly-Alvarado, who had proved this for all but three surfaces. Over general…
Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…
We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the…
We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration…
This is an expanded version of our work [AN88], 1988, in Russian. We classify del Pezzo surfaces over C with log terminal singularities of index \le 2. By classification, we understand a description of the intersection graph of all…
We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.
Sur toute surface de del Pezzo de degr\'e 4 sur un corps $C_1$ de caract\'eristique z\'ero, tous les points rationnels sont R-\'equivalents. Plus g\'en\'eralement, ceci vaut sur tout corps parfait infini de caract\'eristique diff\'erente de…
We consider a real del Pezzo surface without points. We prove that the same surface over complex numbers field $\mathbb{C}$ has Picard number is at least two.
We solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields completely for 85 of the 112 possible types. We also determine for all 112 types the smallest field of existence. As an aside, we provide an example…
We investigate exceptional sheaves on the Hirzebruch surface $\mathbb{F}_2$, as the first attempt toward the classification of exceptional objects on weak del Pezzo surfaces.
In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…
We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…