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Symmetric and alternating groups as monodromy groups of Riemann surfaces I : generic covers and covers with many branch points

Author: Robert M Guralnick; John Shareshian
Publisher: Providence, R.I. : American Mathematical Society, ©2007.
Series: Memoirs of the American Mathematical Society, no. 886.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
The authors consider indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. They show that if the cover has five or more branch points then the genus grows rapidly with $n$ unless either $d = n$ or the curves have genus zero, there are precisely five branch points and $n =d(d-1)/2$. Similarly, if there is a totally ramified point, then without  Read more...
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Document Type: Book
All Authors / Contributors: Robert M Guralnick; John Shareshian
ISBN: 9780821839928 0821839926
OCLC Number: 125406381
Description: vi, 128 pages : illustrations ; 26 cm.
Contents: 1. Introduction and statement of main results --
2. Notation and basic lemmas --
3. Examples --
4. Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 --
5. Actions on $2$-sets--the proof of Theorem 4.0.30 6. --
Actions on $3$-sets--the proof of Theorem 4.0.31 --
7. Nine or more branch points--the proof of Theorem 4.0.34 --8. Actions on cosets of some $2$-homogeneous and $3$-homogeneous groups --
9. Actions on $3$-sets compared to actions on larger sets --
10. A transposition and an $n$-cycle --
11. Asymptotic behavior of $g_k(E)$ --
12. An $n$-cycle--the proof of Theorem 1.2.1 --
13. Galois groups of trinomials--the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3 --
Appendix A. Finding small genus examples by computer search--by R. Guralnick and R. Stafford.
Series Title: Memoirs of the American Mathematical Society, no. 886.
Responsibility: Robert M. Guralnick, John Shareshian ; with an appendix by R. Guralnick and J. Stafford.

Abstract:

Considers indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. The authors show that if the cover has five or more branch points  Read more...

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    schema:description "1. Introduction and statement of main results -- 2. Notation and basic lemmas -- 3. Examples -- 4. Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 -- 5. Actions on $2$-sets--the proof of Theorem 4.0.30 6. -- Actions on $3$-sets--the proof of Theorem 4.0.31 -- 7. Nine or more branch points--the proof of Theorem 4.0.34 --8. Actions on cosets of some $2$-homogeneous and $3$-homogeneous groups -- 9. Actions on $3$-sets compared to actions on larger sets -- 10. A transposition and an $n$-cycle -- 11. Asymptotic behavior of $g_k(E)$ -- 12. An $n$-cycle--the proof of Theorem 1.2.1 -- 13. Galois groups of trinomials--the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3 -- Appendix A. Finding small genus examples by computer search--by R. Guralnick and R. Stafford."@en ;
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