Related papers: Rational points on some Fano cubic bundles

It is shown that hypersurfaces of degree $M$ in ${\mathbb P}^M$, $M\geqslant 5$, with at most quadratic singularities of rank at least 3, satisfying certain conditions of general position, are birationally superrigid Fano varieties and the…

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

For $X$ a smooth cubic threefold we study the Pl\"ucker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a…

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the…

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type…

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the…

We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there…

We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x' \in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\leq 2$. Moreover, a…

We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line…

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by…

We prove that a very general complex hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ containing an $r$-plane with multiplicity $m$ is not stably rational for $n \ge 3$, $m, r > 0$ and $n \ge m+r$. We also investigate failure of stable…

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.

We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show…

We study the spaces of rational curves on Fano threefolds with Gorenstein terminal singularities. We generalize the results regarding Geometric Manin's Conjecture for smooth Fano threefolds, including the classification of subvarieties with…

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular…

In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in…

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…

We test numerically the refined Manin's conjecture about the asymptotics of points of bounded height on Fano varieties for some diagonal cubic surfaces.

In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in…

We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff…