Related papers: Quartic surface, its bitangents and rational point…

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…

Let $S \subset \P^n$ be a smooth quartic hypersurface defined over a number field $K$. If $n \ge 4$, then for some finite extension $K'$ of $K$ the set $S(K')$ of $K'$-rational points of $S$ is Zariski dense.

Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap…

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on…

Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

Let X be a non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to…

We give examples of smooth $\k$-unirational line-free quartic hypersurfaces over a non algebraically closed field $\k$. Unlike other methods of proving unirationality, our method does not rely on existence of linear spaces on quartics.

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of $\mathbb{P}^2$ branched along a smooth sextic curve, we give a bound for the degree…

Let $X$ be a K3 surface defined over a number field $K$. Assume that $X$ admits a structure of an elliptic fibration or an infinite group of automorphisms. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational…

We show that there cannot be more than 64 lines on a quartic surface admitting isolated rational double points over an algebraically closed field of characteristic $p \neq 2,\,3$, thus extending Segre--Rams--Sch\"utt theorem. Our proof…

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition…

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

We show, in this first part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $16$. We produce examples with…

Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense…

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…

Let $k$ be an infinite field of characteristic 0, and $X$ a del Pezzo surface of degree $d$ with at least one $k$-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set…

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We…

Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X,…