Linear series over real and -adic fields

Author:
Brian Osserman

Journal:
Proc. Amer. Math. Soc. **134** (2006), 989-993

MSC (2000):
Primary 14H51, 14P99

DOI:
https://doi.org/10.1090/S0002-9939-05-08247-X

Published electronically:
September 28, 2005

MathSciNet review:
2196029

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Abstract | References | Similar Articles | Additional Information

Abstract: We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

**1.**Siegfried Bosch, Werner Lutkebohmert, and Michel Raynaud,*Neron models*, Springer-Verlag, 1991. MR**1045822 (91i:14034)****2.**David Eisenbud and Joe Harris,*Limit linear series: Basic theory*, Inventiones Mathematicae**85**(1986), 337-371. MR**0846932 (87k:14024)****3.**A. Eremenko and A. Gabrielov,*Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry*, Annals of Mathematics**155**(2002), no. 1, 105-129. MR**1888795 (2003c:58028)****4.**B. Osserman,*Limit linear series in positive characteristic and Frobenius-unstable vector bundles on curves*, Ph.D. thesis, MIT.**5.**-,*The number of linear series on curves with given ramification*, International Mathematics Research Notices**2003**, no. 47, 2513-2527. MR**2007538 (2004g:14011)****6.**J. P. Serre,*Lie algebras and Lie groups*, second ed., Lecture Notes in Mathematics, no. 1500, Springer-Verlag, 1992. MR**1176100 (93h:17001)****7.**Frank Sottile,*The special Schubert calculus is real*, Electronic Research Announcements of the AMS**5**(1999), 35-39. MR**1679451 (2000c:14074)****8.**-,*Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro*, Experimental Mathematics**9**(2000), no. 2, 161-182. MR**1780204 (2001e:14054)****9.**Gayn B. Winters,*On the existence of certain families of curves*, American Journal of Mathematics**96**(1974), no. 2, 215-228. MR**0357406 (50:9874)**

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Additional Information

**Brian Osserman**

Affiliation:
Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840

DOI:
https://doi.org/10.1090/S0002-9939-05-08247-X

Received by editor(s):
March 20, 2004

Received by editor(s) in revised form:
November 7, 2004

Published electronically:
September 28, 2005

Communicated by:
Michael Stillman

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.