THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$
The Hilbert scheme $X^{[3]}$ of length-3 subschemes of a smooth projective variety $X$ is known to be sm...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2016-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509416000256/type/journal_article |
| Summary: | The Hilbert scheme
$X^{[3]}$
of length-3 subschemes of a smooth projective variety
$X$
is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map
$X^{3}{\dashrightarrow}X^{[3]}$
. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.
240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety
$X$
has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers
$X^{n}$
have a filtration, which is the expected Bloch–Beilinson filtration, that is split. |
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| ISSN: | 2050-5094 |