Cocharacter-closure and the rational Hilbert-Mumford Theorem

Michael Bate; Sebastian Herpel; Benjamin Martin; Gerhard Roehrle

doi:10.1007/s00209-016-1816-5

Cocharacter-closure and the rational Hilbert-Mumford Theorem

Michael Bate, Sebastian Herpel, Benjamin Martin, Gerhard Roehrle

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Abstract

For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.

Original language	English
Pages (from-to)	39-72
Number of pages	34
Journal	Mathematische Zeitschrift
Volume	287
Issue number	1-2
Early online date	10 Nov 2016
DOIs	https://doi.org/10.1007/s00209-016-1816-5
Publication status	Published - Oct 2017

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Keywords

math.AG
math.GR
20G15, 14L24
Affine G-variety
Cocharacter-closed orbit
Rationality

Cite this

Bate, M., Herpel, S., Martin, B., & Roehrle, G. (2017). Cocharacter-closure and the rational Hilbert-Mumford Theorem. Mathematische Zeitschrift, 287(1-2), 39-72. https://doi.org/10.1007/s00209-016-1816-5

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abstract = "For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.",

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author = "Michael Bate and Sebastian Herpel and Benjamin Martin and Gerhard Roehrle",

note = "33 pages, v. 2 reference added, slight changes The authors acknowledge the financial support of EPSRC Grant EP/L005328/1 and of Marsden Grants UOC1009 and UOA1021. Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. Also, part of this paper was written during mutual visits to Auckland, Bochum and York. We are grateful to the referees for their careful reading of the paper and for helpful suggestions.",

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AU - Bate, Michael

AU - Herpel, Sebastian

AU - Martin, Benjamin

AU - Roehrle, Gerhard

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N2 - For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.

AB - For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.

KW - math.AG

KW - math.GR

KW - 20G15, 14L24

KW - Affine G-variety

KW - Cocharacter-closed orbit

KW - Rationality

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DO - 10.1007/s00209-016-1816-5

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SP - 39

EP - 72

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

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Cocharacter-closure and the rational Hilbert–Mumford Theorem
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