Lucas Mioranci : Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties

Posted by Kevin Tucker , part of the Algebraic Geometry Seminar.

At

Oct. 23, 2023, 3 p.m.

In

636 SEO

Abstract

A complex projective variety $X$ is algebraically hyperbolic if there exists an ample divisor $H$ and a real number $\epsilon > 0$ such that the geometric genus $g(C)$ and the degree of any integral curve $C\subset X$ satisfy the inequality \[ 2g(C) - 2\ge \epsilon \deg_H (C). \] The algebraic hyperbolicity is an important property to characterize varieties of general type, and it is connected to famous conjectures such as the Lang Conjectures and Green-Griffiths Conjecture. By building on recent work, I classify algebraic hyperbolic hypersurfaces in homogeneous varieties, thus obtaining explicit bounds for the hyperbolicity in plenty of open cases, including Grassmannians, flag varieties, and their products.