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...
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| Format: | Article |
| Language: | English |
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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 |
| _version_ | 1811156248061542400 |
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| author | MINGMIN SHEN CHARLES VIAL |
| author_facet | MINGMIN SHEN CHARLES VIAL |
| author_sort | MINGMIN SHEN |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T04:47:20Z |
| format | Article |
| id | doaj.art-98ccce1346ec47a296e0aa6856b3048b |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:20Z |
| publishDate | 2016-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-98ccce1346ec47a296e0aa6856b3048b2023-03-09T12:34:41ZengCambridge University PressForum of Mathematics, Sigma2050-50942016-01-01410.1017/fms.2016.25THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$MINGMIN SHEN0CHARLES VIAL1KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands;DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK;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.https://www.cambridge.org/core/product/identifier/S2050509416000256/type/journal_article14C05 (primary)14C2514C15 (secondary) |
| spellingShingle | MINGMIN SHEN CHARLES VIAL THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ Forum of Mathematics, Sigma 14C05 (primary) 14C25 14C15 (secondary) |
| title | THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ |
| title_full | THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ |
| title_fullStr | THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ |
| title_full_unstemmed | THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ |
| title_short | THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ |
| title_sort | motive of the hilbert cube x 3 |
| topic | 14C05 (primary) 14C25 14C15 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509416000256/type/journal_article |
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