Topics in the Theory of Riemann SurfacesThe book's main concern is automorphisms of Riemann surfaces, giving a foundational treatment from the point of view of Galois coverings, and treating the problem of the largest automorphism group for a Riemann surface of a given genus. In addition, the extent to which fixed points of automorphisms are generalized Weierstrass points is considered. The extremely useful inequality of Castelnuovo- Severi is also treated. While the methods are elementary, much of the material does not appear in the current texts on Riemann surfaces, algebraic curves. The book is accessible to a reader who has had an introductory course on the theory of Riemann surfaces or algebraic curves. |
Contents
| 1 | |
Some exceptional points on Riemann surfaces | 13 |
The inequality of CastelnuovoSeveri 20 | 20 |
Smooth and branched coverings of Riemann surfaces | 30 |
Automorphisms of Riemann surfaces I | 42 |
| 99 | |
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Common terms and phrases
A₁ abelian Algebraic automorphism groups automorphism of order base points big group biholomorphic branch points called Castelnuovo-Severi Clifford's theorem compact Riemann surfaces complete linear series composition series corresponding cover transformation cyclic group defined denoted dihedral group distinct points elliptic-hyperelliptic equivalent finite group fixed points follows function field Galois covering group of automorphisms group of genus group of order holomorphic map homeomorphism hyperelliptic involution hyperelliptic Riemann surface integer integral divisor invariant involution isomorphic Lemma Let f Let G lifts line bundle meromorphic differential meromorphic function modn monodromy n-sheeted no(G non-gaps normal subgroup parameter permutation poles positive integer prime integer Proceedings proof of Theorem ramification Riemann-Hurwitz formula Riemann-Roch theorem singularities smooth covering Suppose G surface of genus tacnodes Theory VIII Weierstrass points x₁ y₁ Z₂ α α αβ