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Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves

Part of: General commutative ring theory Birational geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 September 2008

Burt Totaro
Burt Totaro*
Affiliation:
DPMMS, Wilberforce Road, Cambridge CB3 0WB, UK (email: b.totaro@dpmms.cam.ac.uk)
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Abstract

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We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

Keywords

finite generationring of invariantselliptic fibrationCalabi–Yau fiber spaceklt paircone conjecture

MSC classification

Secondary: 13A50: Actions of groups on commutative rings; invariant theory 14E30: Minimal model program (Mori theory, extremal rays) 14J32: Calabi-Yau manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 5 , September 2008 , pp. 1176 - 1198
Copyright
Copyright © Foundation Compositio Mathematica 2008