Related papers: Quartic K3 surfaces without nontrivial automorphis…

K3 surfaces with non-symplectic involution are classified by open sets of seventy-five arithmetic quotients of type IV. We prove that those moduli spaces are rational except two classical cases.

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the…

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ on a smooth Hermitian surface $X$ of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$. We prove that $R$ is the…

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…

Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C} \cap \mathcal{Q}$ is even and $\mathcal{C}…

Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point…

We introduce Kummer surfaces X=Km(CxC) with the group scheme G=mu_2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational…

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.

We study points and 0-cycles on del Pezzo surfaces defined over a field K of characteristic 0, with emphasis on cubic surfaces. We prove that a cubic surface that admits a point defined over a field extension of K of degree coprime to 3…

We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the…

In each characteristic $p\neq 2, 3$, it was shown in a previous work that the order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it is shown that in each characteristic $p\neq 2, 3$ a K3 surface with a cyclic action…

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field.…

We give a counterexample to the following conjecture: the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. As a positive result, we prove that any cohomologically…

We prove the sharp bound of at most 64 lines on complex projective quartic surfaces (resp. affine quartics) that are not ruled by lines. We study configurations of lines on certain non-K3 surfaces of degree four and give various examples of…

We classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and points and we completely classify the seven possible configurations. If the…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}_\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that…

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of…

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…