Subvarieties of complete intersections of large degree

TL;DR

This paper investigates the properties of subvarieties within very general complete intersections of large degree in projective space, establishing optimal bounds for when these subvarieties are of general type and characterizing hyperbolicity.

Contribution

It improves existing bounds for subvarieties of complete intersections to be of general type and characterizes algebraic hyperbolicity for certain cases, extending previous results with new techniques.

Findings

Optimal bound for subvarieties to be of general type: d ≥ 2n - c - k

Characterization of algebraic hyperbolicity for certain complete intersections

Identification of loci where non-general type curves and positive-dimensional orbits lie

Abstract

We study subvarieties of very general complete intersections $X\subset \mathbb{P}^n$ of multidegree $(d_1,\dots,d_c)$, when $d:= d_1+\dots +d_c$ is sufficiently large. In a seminal paper Ein proved that if $d\geq 2n-c-k+2$, any $k$-dimensional subvariety of $X$ is of general type and has positive geometric genus. We strengthen this result by obtaining the optimal bound $d\geq 2n-c-k$, provided that $n> 2c+k$. As a consequence, we characterize algebraic hyperbolicity of very general complete intersections $X\subset \mathbb{P}^n$ of codimension $c\leq \frac{n-3}{2}$. For lower values of $d$, we prove that if $\frac{3n-c+2}{2}\leq d\leq 2n-c-2$ and $(d_1,\dots,d_c)$ satisfies an additional numerical condition, then the only curves in $X$ that are not of general type are lines. Moreover, we describe the locus where positive dimensional orbits of points under rational equivalence must lie. We obtain our results by proving that, under suitable numerical conditions, subvarieties of $X$ that are not of general type must lie in the locus of $X$ covered by lines. The proof of this result relies on a generalization of the approach and techniques developed for hypersurfaces by Voisin, Clemens-Ran and the second author, combined with a Grassmannian technique introduced by Riedl-Yang.