Criteria for σ-ampleness

Abstract

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.

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

Cited by in corpus (10)

• Naive Noncommutative Blowing Up
• Ample filters and Frobenius amplitude
• Geometric idealizers
• Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings
• Geometric algebras on projective surfaces
• Wild automorphisms of projective varieties, the maps which have no invariant proper subsets
• Noncommutative ampleness for multiple divisors
• Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)
• An upper bound for polynomial volume growth of automorphisms of zero entropy
• Noncommutative ampleness from finite endomorphisms