# 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
22 pages

/lp/de-gruyter/divisors-on-real-curves-vcKUuGt0gS
Publisher
de Gruyter
ISSN
1615-715X
DOI
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

Published: Aug 19, 2003

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations