Regular homomorphisms, with a twist
Jeff Achter, Sebastian Casalaina-Martin, Charles Vial

TL;DR
This paper introduces a new perspective on regular homomorphisms by associating torsors over an abelian variety to certain cycle data, providing obstructions to algebraic cycle existence and linking to rationality problems of threefolds.
Contribution
It demonstrates that cycle data without $K$-points determines torsors over $K$, offering a novel obstruction to algebraic cycle existence and connecting to recent rationality results.
Findings
Cycle data without $K$-points defines torsors over $K$.
Obstruction to algebraic cycle existence over $K$.
Links to rationality of threefolds via recent results.
Abstract
Let be a variety over a field, and an abelian variety. A regular homomorphism to (in codimension ) induces, for every smooth geometrically connected pointed -scheme and every cycle class , a morphism of varieties over . In this note we show that, if admits no -point, the data determines a torsor over under and a -morphism . This can be used to provide an obstruction to the existence of algebraic cycles defined over . We then connect this obstruction to some recent results of Hassett--Tschinkel and Benoist--Wittenberg on rationality of threefolds.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Commutative Algebra and Its Applications
