Related papers: Intermediate Jacobians and rationality over arbitr…

Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of…

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.

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.

Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…

We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that…

The purpose of this article is to initiate a study of a class of Lorentz invariant, yet tractable, Lagrangian Field Theories which may be viewed as an extension of the Klein-Gordon Lagrangian to many scalar fields in a novel manner. These…

Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction…

Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that…

In this article, we determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces, one contains a…

We show that there exists a $2$-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This together with Clemens--Griffiths' work solves the long standing open problem whether there exists…

Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp.…

We prove that for any rationally connected threefold $X$, there exists a smooth projective surface $S$ and a family of $1$-cycles on $X$ parameterized by $S$, inducing an Abel-Jacobi isomorphism ${\rm Alb}(S)\cong J^3(X)$. This statement…

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold $X$ containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to $X$ up to a…

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…

Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times…

We provide new logarithmic lower bounds for the torsion order of a very general complete intersection in projective space as well as a very general hypersurface in products of projective spaces and Grassmannians, in particular we prove…

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N=2g^2+7g+1$, we introduce a…

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…