Difference between revisions of "Rational curve"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Fulton, "Algebraic curves" , Benjamin (1969) pp. 66 {{MR|0313252}} {{MR|0260752}} {{ZBL|0194.21901}} {{ZBL|0181.23901}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table> |
Revision as of 21:55, 30 March 2012
A one-dimensional algebraic variety, defined over an algebraically closed field 
, whose field of rational functions is a purely transcendental extension of degree 1 of 
. Every non-singular complete rational curve is isomorphic to the projective line 
. A complete singular curve 
is rational if and only if its geometric genus 
is zero, that is, when there are no regular differential forms on 
.
When 
is the field 
of complex numbers, the (only) non-singular complete rational curve 
is the Riemann sphere 
.
Comments
In classic literature a rational curve is also called a unicursal curve.
If 
is defined over a not necessarily algebraically closed field 
and 
is birationally equivalent to 
over 
, 
is said to be a 
-rational curve.
References
| [a1] | W. Fulton, "Algebraic curves" , Benjamin (1969) pp. 66 MR0313252 MR0260752 Zbl 0194.21901 Zbl 0181.23901 |
| [a2] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
How to Cite This Entry:
Rational curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_curve&oldid=23948
Rational curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_curve&oldid=23948
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article