Criteria for σ-ampleness

Details Criteria for σ-ampleness Criteria for σ-ampleness

1999-12-06

arxiv.org/abs/math/9912051v2

J. Amer. Math. Soc. 13 (2000), no. 3, 517-532.

Mathematics

Algebraic Geometry

16 pages, LaTeX2e, to appear in J. of the AMS, minor errors corrected (esp. in 1.4 and 3.1), proofs simplified

Scientific paper

10.1090/S0894-0347-00-00334-9

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism of a projective scheme X. Many open questions regarding $\sigma$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $\sigma$-ample. As a consequence, we show right and left $\sigma$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.

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