1-Cycles on Fano varieties

TL;DR

This paper investigates the structure of 1-cycles on specific Fano varieties, establishing conditions under which their Chow groups are trivial or generated by lines, using rational curve techniques.

Contribution

It provides new results on the Chow groups of Fano weighted complete intersections and conic-connected varieties, highlighting cases with trivial or line-generated first Chow groups.

Findings

Fano weighted complete intersections with high index have trivial first Griffiths group.

Most 2-Fano weighted complete intersections have their first Chow group generated by lines.

If the Fano variety of lines is irreducible, the first Chow group is isomorphic to .

Abstract

We prove results about 1-cycles on certain Fano varieties using techniques that rely on rational curves. Firstly, we show that Fano weighted complete intersections with index bigger than half their dimension have trivial first Griffiths group. Secondly, we prove that the first Chow group of most $2$-Fano weighted complete intersections, and of $2$-Fano conic-connected varieties in $\mathbb{P}^n$ of high enough index (with $3$ obvious exceptions), are generated by lines. Furthermore, if the Fano variety of lines is irreducible, the first Chow group is isomorphic to $\mathbb{Z}$.