English

Subvarieties of complete intersections of large degree

Algebraic Geometry 2026-02-16 v1

Abstract

We study subvarieties of very general complete intersections XPnX\subset \mathbb{P}^n of multidegree (d1,,dc)(d_1,\dots,d_c), when d:=d1++dcd:= d_1+\dots +d_c is sufficiently large. In a seminal paper Ein proved that if d2nck+2d\geq 2n-c-k+2, any kk-dimensional subvariety of XX is of general type and has positive geometric genus. We strengthen this result by obtaining the optimal bound d2nckd\geq 2n-c-k, provided that n>2c+kn> 2c+k. As a consequence, we characterize algebraic hyperbolicity of very general complete intersections XPnX\subset \mathbb{P}^n of codimension cn32c\leq \frac{n-3}{2}. For lower values of dd, we prove that if 3nc+22d2nc2\frac{3n-c+2}{2}\leq d\leq 2n-c-2 and (d1,,dc)(d_1,\dots,d_c) satisfies an additional numerical condition, then the only curves in XX 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 XX that are not of general type must lie in the locus of XX 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.

Keywords

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