Subvarieties of complete intersections of large degree
Abstract
We study subvarieties of very general complete intersections of multidegree , when is sufficiently large. In a seminal paper Ein proved that if , any -dimensional subvariety of is of general type and has positive geometric genus. We strengthen this result by obtaining the optimal bound , provided that . As a consequence, we characterize algebraic hyperbolicity of very general complete intersections of codimension . For lower values of , we prove that if and satisfies an additional numerical condition, then the only curves in 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 that are not of general type must lie in the locus of 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.
Cite
@article{arxiv.2602.13083,
title = {Subvarieties of complete intersections of large degree},
author = {Francesco Bastianelli and Gianluca Pacienza},
journal= {arXiv preprint arXiv:2602.13083},
year = {2026}
}
Comments
37 pages. Comments are welcome