Abstract
A surface X is a ruled surface if there exists a nonsingular projective curve B such that X is birationally isomorphic to P1 × B.
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Bibliographic References
The proof of Theorem 11.6 (which is essential for the proof of the Noether-Tsen criterion) is taken from [Burl].
Theorem 11.10 is a variant of the Noether-Tsen criterion; this variant, via its Corollary 11.11, makes the connection between geometrically ruled surfaces and rank 2 vector bundles on nonsingular projective curves.
The facts regarding vector bundles on curves are presented after [Ati], and the simple proof of Grothendiecks theorem in the special case of vector bundles of rank 2, cf. Theorem 11.22, is from [Bea].
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© 2001 Springer Science+Business Media New York
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Bădescu, L. (2001). Ruled Surfaces. The Noether-Tsen Criterion. In: Algebraic Surfaces. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3512-3_11
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DOI: https://doi.org/10.1007/978-1-4757-3512-3_11
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