Related papers: Rationality of singular cubic threefolds over $\ma…

In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group…

We study unirationality of actions of finite groups on Fano threefolds.

Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

This short note is an extended abstract of a talk given at the conference "Komplexe Analysis" at the Mathematisches Forschungsinstitut Oberwolfach in September 2012. We explained some recent results about the existence of rational curves on…

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

Any smooth projective curve embeds into $\mathbb{P}^3$. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold $X$, then $X$ is…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…

We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.

We determine all the Kummer-surface-type Calabi-Yau (CY) 3-folds, i.e., those $\hat{T/G}$ which are resolutions of 3-torus-orbifolds $T/G$ with only isolated singularities. There are only two such CY spaces: one with $G= \ZZ_3$ and $T$…

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…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

We study finite-dimensional spaces of rational one-forms on a projective manifold by means of their integrable locus.

In this paper we classify three-dimensional singular cubic hypersurfaces with an action of a finite group $G$, which are not $G$-rational, are not $G$-birationally isomorphic to a quadric and have no birational structure of $G$-Mori fiber…

We prove rationality of the moduli variety of curves of genus 3.

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 smooth the singularities of a strictly hyperbolized smooth cube manifold.

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this…

We classify special self-birational transformations of the smooth quadric threefold and fourfold, $Q^3$ and $Q^4$. It turns out that there is only one such example in each dimension. In the case of $Q^3$, it is given by the linear system of…

A rational map between certain specific threefolds is given in an explicit manner.