Torsion orders of complete intersections

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.