We construct invariants of birational maps with values in the Kontsevich--Tschinkel group
and in the truncated Grothendieck groups of varieties. These invariants are morphisms of …

This paper is inspired by the discovery by Larsen and Lunts [10] of a remarkable connection
between motivic integration and stable rationality and by the recent development of these …

We prove that stable rationality specializes in regular families whose fibers are integral and
have at most ordinary double points as singularities. Our proof is based on motivic …

We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic
hypersurface, or equivalently, the universal triviality of its CH0 group. We prove that for odd …

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality
questions. Topics include classical constructions of rational examples, Hodge structures and …

We discuss a conjecture saying that derived equivalence of smooth projective simply
connected varieties implies that the difference of their classes in the Grothendieck ring of …

We present a new tool for the calculation of Denef and Loeser's motivic nearby fiber and
motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on …

The circle method has been successfully used over the last century to study rational points
on hypersurfaces. More recently, a version of the method over function fields, combined with …

The class of the affine line is a zero divisor in the Grothendieck ring: An improvement -
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One of the simplest and, at the same time, most important invariants of a topological space is
the Euler characteristic. A generalization of the notion of the Euler characteristic to the …