Divisors on real curves

Divisors on real curves Abstract. Let X be a smooth projective curve over R. In the first part, we study e¤ective divisors on X with totally real or totally complex support. We give some numerical conditions for a linear system to contain such a divisor. In the second part, we describe the special linear systems on a real hyperelliptic curve and prove a new Cli¤ord inequality for such curves. Finally, we study the existence of complete linear systems of small degrees and dimension r on a real curve. 2000 Mathematics Subject Classification. 14C20, 14H51, 14P25, 14P99 Introduction In this note, a real algebraic curve X is a smooth proper geometrically integral scheme over R of dimension 1. A closed point P of X will be called a real point if the residue field at P is R, and a non-real point if the residue field at P is C. The set of real points, X ðRÞ, will always be assumed to be non-empty. It decomposes into finitely many connected components, whose number will be denoted by s. By Harnack's theorem we know that 1 c s c g þ 1, where g is the genus of X . A curve with g http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

Divisors on real curves

Advances in Geometry, Volume 3 (3) – Aug 19, 2003

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Publisher
de Gruyter
Copyright
Copyright © 2003 by the
ISSN
1615-715X
DOI
10.1515/advg.2003.019
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let X be a smooth projective curve over R. In the first part, we study e¤ective divisors on X with totally real or totally complex support. We give some numerical conditions for a linear system to contain such a divisor. In the second part, we describe the special linear systems on a real hyperelliptic curve and prove a new Cli¤ord inequality for such curves. Finally, we study the existence of complete linear systems of small degrees and dimension r on a real curve. 2000 Mathematics Subject Classification. 14C20, 14H51, 14P25, 14P99 Introduction In this note, a real algebraic curve X is a smooth proper geometrically integral scheme over R of dimension 1. A closed point P of X will be called a real point if the residue field at P is R, and a non-real point if the residue field at P is C. The set of real points, X ðRÞ, will always be assumed to be non-empty. It decomposes into finitely many connected components, whose number will be denoted by s. By Harnack's theorem we know that 1 c s c g þ 1, where g is the genus of X . A curve with g

Journal

Advances in Geometryde Gruyter

Published: Aug 19, 2003

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