A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

TL;DR

This paper proves a Noether-Lefschetz type theorem for the Picard number of varieties of r-planes in general complete intersections, showing it is 1 except in specific low-dimensional quadric cases, with applications to cubic fivefolds.

Contribution

It establishes a new Picard number result for varieties of r-planes in complete intersections, extending Noether-Lefschetz theory to these geometric objects.

Findings

Picard number of F_r(X) is 1 for general complete intersections when dimension is at least 2

Exceptions occur for certain low-dimensional quadrics and intersections of quadrics

Application to cohomology class determination of planes in cubic fivefolds

Abstract

Let X be a very general complete intersection in complex projective space and we denote by $F_r(X)$ the variety of r-planes in X, for $r\geq 1$. We show that the Picard number of $F_r(X)$ is 1, as soon as $\dim F_r(X)\geq 2$, except when X is a quadric of dimension 2r or 2r+2, or X is a complete intersection of two quadrics of dimension 2r+2. We also apply this result to determine the cohomology class of the variety of planes of a cubic fivefold contained (by the Abel-Jacobi map) in the intermediate Jacobian.