Torsion orders of complete intersections

Abstract

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