Related papers: On smooth rational threefolds of P^5 with rational…

In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all…

We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to…

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian…

The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a…

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 the case of a $K3$ surface of Picard rank two such that…

This paper presents new examples of projective surfaces of general type over $\mathbb{C}$ with canonical map of degree $ 3 $ onto a surface of general type. Very few examples are known of such surfaces and some of the examples in this paper…

We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some…

We exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.

Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by…

We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction,…

The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of…

We study rational surfaces on very general Fano hypersurfaces in $\mathbb{P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree $d$ sufficiently close to…

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections $S\hookrightarrow \mathbb P^g\dashrightarrow \mathbb P^2$ of minimal degree and their variational structure. In particular, we prove that…

Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt…

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has…

In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up $\mathbb…

In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has…

We prove non-rationality and birational super-rigidity of a Q-factorial double cover X of P^3 ramified along a sextic surface with at most simple double points. We also show that the condition #|Sing(X)| < 15 implies Q-factoriality of X. In…

We prove that there exist smooth surfaces of degree d in projective 3-space such that the group of rational equivalence classes of decomposable 0-cycles has rank at least the integer part of (d-1)/3.