A note on Positivity of the CM line bundle

Details A note on Positivity of the CM line bundle A note on Positivity of the CM line bundle

2006-05-11

arxiv.org/abs/math/0605302v2

Mathematics

Algebraic Geometry

10 pages; To appear in IMRN. Corrected typos and the statement of Thoerem 4.2 following referee's comments

Scientific paper

We show that positivity of the CM line associated to a family of polarised varieties is intimately related to the stability of its members. We prove that the CM line is nef on any curve which meets the stable locus, and that it is pseudoeffective (i.e. in the closure of the effective cone) as long as there is at least one stable fibre. We give examples showing that the CM line can be strictly negative or strictly positive on curves in the unstable locus.

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