A remark on the motive of the Fano variety of lines of a cubic
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remark on the motive of the Fano variety of lines of a cubic Robert Laterveer1
Received: 26 February 2016 / Accepted: 3 October 2016 / Published online: 22 October 2016 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract Let X be a smooth cubic hypersurface, and let F be the Fano variety of lines on X . We establish a relation between the Chow motives of X and F. This relation implies in particular that if X has finite-dimensional motive (in the sense of Kimura), then F also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension 3 and 5, and of certain cubics in other dimensions. Keywords Algebraic cycles · Chow groups · Motives · Finite-dimensional motives · Cubics · Fano variety of lines Résumé Soit X une hypersurface cubique lisse, et soit F la variété de Fano paramétrant les droites contenues dans X . On établit une relation entre les motifs de Chow de X et de F. Cette relation implique le fait que F a motif de dimension finie (au sens de Kimura) à condition que X a motif de dimension finie. En particulier, si X est une cubique lisse de dimension 3 ou 5, alors F a motif de dimension finie. Mathematics Subject Classification 14C15 · 14C25 · 14C30 · 14J70 · 14N25
1 Introduction The notion of finite-dimensional motive, developed independently by Kimura and O’Sullivan [1,10,14,15,19] has given important new impetus to the study of algebraic cycles. To give but one example: thanks to this notion, we now know the Bloch conjecture is true for surfaces of geometric genus zero that are rationally dominated by a product of curves [15]. It thus seems worthwhile to find concrete examples of varieties that have finite-dimensional motive, this being (at present) one of the sole means of arriving at a satisfactory understanding of Chow groups.
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Robert Laterveer [email protected] CNRS-IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg cedex, France
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The present note aims to contribute something to the list of examples of varieties with finite-dimensional motive, by considering Fano varieties of lines of smooth cubics over C. The main result is as follows: Theorem (=Theorem 4) Let X ⊂ Pn+1 (C) be a smooth cubic hypersurface, and let F(X ) denote the Fano variety of lines on X . If X has finite-dimensional motive, then also F(X ) has finite-dimensional motive. In particular, this implies that for smooth cubics X of dimension 3 or 5, the Fano variety F(X ) has finite-dimensional motive. In the first case, the dimension of F(X ) is 2, while in the second case it is 6. The case n = 3 is also proven (in a different way) in [5]. Some more examples where Theorem 4 applies are given in Corollary 17. Theorem 4 follows from a more general result. This more general result relates the Chow motives of X and F = F(X ) for any smooth cubic: Theorem (=Theorem 5) Let X ⊂ Pn+1 (C) be a smooth cubic hypersurface. Let F := F(X ) denote the Fano variety of lines on X , and let X [2] denote the sec
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