Subvarieties of complete intersections of large degree

TL;DR

The paper tackles subvarieties of very general complete intersections $X\subset \mathbb{P}^n$ with large total degree, establishing the optimal bound $d \ge 2n-c-k$ (when $n>2c+k$) that forces $k$-dimensional subvarieties to be of general type (or to have positive genus). It extends hypersurface techniques to arbitrary codimension via a new framework of vertical tangent spaces, (bi)contact loci, and a Grassmannian method, proving hyperbolicity criteria for codimension up to $(n-3)/2$ and isolating lines as the only non-general-type curves in certain low-degree ranges. The core strategy binds the geometry of contact loci to integrable foliations in the vertical direction and uses multiplication maps to constrain base loci, then leverages Grassmannian arguments to rule out unwanted subvarieties outside the lines locus. Consequently, the authors derive a comprehensive picture linking hyperbolicity, the presence of lines, and rational equivalence orbits, with broad implications for the geometry of complete intersections. The results sharpen Ein’s bounds, extend Voisin–Clemens–Ran techniques to higher codimension, and yield several corollaries on hyperbolicity and conic/line geometry in very general complete intersections.

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.

Theorems & Definitions (55)

• Theorem A
• Corollary B
• Theorem C
• Theorem D
• Theorem E
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: see e.g. P03