Algebraic hyperbolicity of subvarieties of homogeneous varieties

TL;DR

This work addresses the problem of algebraic hyperbolicity for subvarieties of homogeneous varieties, extending established hypersurface results to higher codimension. It develops a framework using section-dominating line bundles, Lazarsfeld-Mukai bundles, and an advanced scroll/cone join construction to bound the genus of curves on a very general complete intersection $X$ within a homogeneous ambient $A$, yielding concrete degree conditions. The main contributions include sufficient criteria guaranteeing hyperbolicity (via a curve-type genus bound and a refined scroll argument) and a complementary low-degree non-hyperbolicity bound; these are then specialized to the complete intersection case to obtain explicit thresholds, notably the sharp bound for hypersurfaces in projective space: $\sum_j d_j \ge 2n-k$ for hyperbolicity and $\sum_j d_j \le 2n-k-2$ for non-hyperbolicity when $k\le n-2$. Collectively, the results generalize Ein, Yeong, Mioranci, and related work to higher codimension subvarieties in homogeneous varieties, providing a robust toolkit (scrolls, joins, and LM-bundles) for analyzing curves on these varieties.

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 $\sum d_j\le 2n-k-2$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4
• Definition 2.1
• Example 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Lemma 2.6