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Difference between revisions of "Rational curve"

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A one-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775801.png" />, whose field of rational functions is a purely [[Transcendental extension|transcendental extension]] of degree 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775802.png" />. Every non-singular complete rational curve is isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775803.png" />. A complete singular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775804.png" /> is rational if and only if its geometric genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775805.png" /> is zero, that is, when there are no regular differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775806.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775807.png" /> is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775808.png" /> of complex numbers, the (only) non-singular complete rational curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r0775809.png" /> is the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758010.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758011.png" /> is defined over a not necessarily algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758013.png" /> is birationally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758016.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077580/r07758018.png" />-rational curve.
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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
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article