Related papers: Algebraic hyperbolicity of surfaces in Fano threef…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times…

We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely…

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on…

We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective $3$-space to have Picard number $>1$. These algebraic surfaces are of general type. We do this by considering degenerations (along an edge),…

We study Fano threefolds with Picard number one equipped with a holomorphic section in $\Omega_V^1(1)$.

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.

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the…

A Fano variety of Picard number $1$ is said to be \textit{birationally solid} if it is not birational to a Mori fiber space over a positive dimensional base. In this paper we complete the classification of quasi-smooth birationally solid…

We show that a non-toric $\mathbb{Q}$-factorial terminal Fano threefold of Picard rank $1$ and Fano index $13$ is a weighted hypersurface of degree $12$ in $\mathbb{P}(3,4,5,6,7)$.

We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations…

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very…

Fano surfaces parametrize the lines of smooth cubic threefolds. In this paper, we study their quotients by some of their automorphism sub-groups. We obtain in that way some interesting surfaces of general type.

We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane…

We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in rational homogeneous varieties. As examples, we work out the cases of…

We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero.

We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This…

We study weighted Fano fourfolds of K3 type realized as hypersurfaces in weighted projective spaces. Under the additional assumption that the singular locus has dimension at most one, we prove that only finitely many such families exist. We…