Torsion orders of complete intersections

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-Thélène and Pirutka. We give a lower bound for the generic complete intersection with and without point. Moreover, we use methods of Kollár and Totaro to show lower bounds for the very general complete intersection.

50 pages. A reference to the paper by Bruno Kahn, "Torsion order of smooth projective surfaces, with an appendix by J.L. Colliot-Thélène" arXiv:1605.01762 [math.AG], was added and some typos corrected. Fixed a gap in the proof of the second main theorem. The statement of the second main theorem was modified in order to exclude the odd dimensional case when the prime 2 is considered

References in corpus (1)

• On stable rationality of Fano threefolds and del Pezzo fibrations

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• Torsion orders of Fano hypersurfaces
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• On the rationality problem for low degree hypersurfaces
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