Difference between revisions of "Rational curve"
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| − | A one-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field | + | {{TEX|done}} |
| + | A one-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field $k$, whose field of rational functions is a purely [[Transcendental extension|transcendental extension]] of degree 1 of $k$. Every non-singular complete rational curve is isomorphic to the projective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no regular differential forms on $X$. | ||
| − | When | + | When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere $\mathbf C\cup\{\infty\}$. |
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In classic literature a rational curve is also called a unicursal curve. | In classic literature a rational curve is also called a unicursal curve. | ||
| − | If | + | If $X$ is defined over a not necessarily algebraically closed field $k$ and $X$ is birationally equivalent to $P_k^1$ over $k$, $X$ is said to be a $k$-rational curve. |
====References==== | ====References==== | ||
<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> | <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 10:10, 15 April 2014
A one-dimensional algebraic variety, defined over an algebraically closed field $k$, whose field of rational functions is a purely transcendental extension of degree 1 of $k$. Every non-singular complete rational curve is isomorphic to the projective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no regular differential forms on $X$.
When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere $\mathbf C\cup\{\infty\}$.
Comments
In classic literature a rational curve is also called a unicursal curve.
If $X$ is defined over a not necessarily algebraically closed field $k$ and $X$ is birationally equivalent to $P_k^1$ over $k$, $X$ is said to be a $k$-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