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Abstract Let $$\alpha \colon X \to Y$$ be a general degree $$r$$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $$r$$. We prove that the Tschirnhausen bundle of $$\alpha $$ is semistable if $$g(Y) \geq 1$$ and stable if $$g(Y) \geq 2$$.
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Projectivity of the moduli of curves
In this expository paper, we show that the Deligne–Mumford moduli space of stable curves is projective over Spec(Z). The proof we present is due to Kollár. Ampleness of a line bundle is deduced from the nefness of a related vector bundle via the ampleness lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve.
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- Award ID(s):
- 1902616
- PAR ID:
- 10585475
- Editor(s):
- Belmans, Pieter; Ho, Wei; de_Jong, Aise Johan
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- ISBN:
- 9781009054850
- Page Range / eLocation ID:
- 1-43
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Book Chapter:
- https://doi.org/10.1017/9781009051897.003
