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The global geometry of the moduli space of curves
This is a survey written for the Proceedings of the AMS Summer Institute in
Algebraic Geometry held in Seattle in 2005. Topics discussed in the survey
include the ample and the effective cone of the moduli space of curves, Kodaira
dimension, Slope Conjecture, log canonical models etc.Comment: 23 pages. Minor revisions. To appear in the Proceedings of the AMS
Summer Research Institute in Algebraic Geometry-Seattle 200
The Geometry of the Moduli Space of Curves of Genus 23
We prove that the Kodaira dimension of the moduli space M_{23} of curves of
genus 23 is at least 2. We also present some evidence for the hypothesis that
the Kodaira dimension of the moduli space is actually equal to 2. Note that for
g > 23 the moduli space is of general type, while for g\leq 22, Harris and
Morrison conjectured that M_g is uniruled. The result on M_{23} is obtained by
investigating the relative position of three explicit multicanonical divisors
which are of Brill-Noether type.Comment: 21 pages, Late
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