Algebraic hyperbolicity of subvarieties of homogeneous varieties

TL;DR

This paper investigates the algebraic hyperbolicity of subvarieties within homogeneous varieties, extending previous results from hypersurfaces to higher codimension complete intersections using advanced algebraic geometry techniques.

Contribution

It generalizes known hyperbolicity criteria from hypersurfaces to higher codimension subvarieties in projective spaces, providing new degree bounds for hyperbolicity.

Findings

Complete intersections with sum of degrees ≥ 2n - k are algebraically hyperbolic.

Complete intersections with sum of degrees ≤ 2n - k - 2 are not algebraically hyperbolic.

Extends techniques of Coskun-Riedl, Yeong, and Mioranci to broader classes of subvarieties.

Abstract

We study the algebraic hyperbolicity of certain subvarieties of homogeneous varieties, building on the techniques introduced by Coskun-Riedl, Yeong and Mioranci. This generalizes earlier known results for hypersurfaces to higher codimensions. In particular, we observe that if $X=X_1\cap\cdots\cap X_k$ is a very general complete intersection of degree $d_j$ hypersurfaces $X_j$ in $\mathbb{P}^n$ with $k\leq n-2$, then $X$ is algebraically hyperbolic if $\sum d_j\ge 2n-k$, and $X$ is not algebraically hyperbolic if $\sum d_j\le 2n-k-2$.