Torsion orders of complete intersections

TL;DR

This paper investigates the torsion order of complete intersections, establishing lower bounds using specialization and algebraic geometry techniques, advancing understanding of rational decompositions of the diagonal.

Contribution

It provides new lower bounds for the torsion order of complete intersections, both generic and very general, employing specialization and Kollár-Totaro methods.

Findings

Lower bounds for torsion order of generic complete intersections.

Lower bounds for torsion order of very general complete intersections.

Application of specialization, Kollár, and Totaro methods to study rational decompositions.

Abstract

By a classical result of Roitman, a complete intersection $X$ of sufficiently small degree admits a rational decomposition of the diagonal. This means that some multiple of the diagonal by a positive integer $N$, when viewed as a cycle in the Chow group, has support in $X\times D\cup F\times X$, for some divisor $D$ and a finite set of closed points $F$. The minimal such $N$ is called the torsion order. We study lower bounds for the torsion order following the specialization method of Voisin, Colliot-Th\'el\`ene and Pirutka. We give a lower bound for the generic complete intersection with and without point. Moreover, we use methods of Koll\'ar and Totaro to show lower bounds for the very general complete intersection.