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Difference between revisions of "Unirational variety"

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An [[Algebraic variety|algebraic variety]] $X$ over a field $k$ into which there exists a rational mapping from a projective space, $\phi : \mathbf{P}^n \rightarrow X$, such that $\phi(\mathbf{P}^n)$ is dense in $X$ and the extension of the field of rational functions $k(\mathbf{P}^n)/k(X)$ is separable. In other words, $k(X)$ has a separable extension which is purely transcendental (cf. [[Transcendental extension|Transcendental extension]]).
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An [[Algebraic variety|algebraic variety]] $X$ over a field $k$ into which there exists a [[rational mapping]] from a projective space, $\phi : \mathbf{P}^n \rightarrow X$, such that $\phi(\mathbf{P}^n)$ is dense in $X$ and the extension of the field of rational functions $k(\mathbf{P}^n)/k(X)$ is separable. In other words, $k(X)$ has a [[separable extension]] which is purely transcendental (cf. [[Transcendental extension]]).
  
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Unirational varieties are close to rational varieties (cf. [[Rational variety|Rational variety]]), e.g. on a unirational variety there are no regular differential forms, $H^0(X,\Omega_X^p) = 0$ for $p \ge 1$. The problem of the coincidence of rational and unirational varieties is called the [[Lüroth problem|Lüroth problem]]; the answer is negative.
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Unirational varieties are close to rational varieties (cf. [[Rational variety]]), e.g. on a unirational variety there are no regular [[differential form]]s, $H^0(X,\Omega_X^p) = 0$ for $p \ge 1$. The problem of the coincidence of rational and unirational varieties is called the [[Lüroth problem]]; the answer is negative.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
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</table>
  
 
[[Category:Algebraic geometry]]
 
[[Category:Algebraic geometry]]
  
 
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Revision as of 10:17, 2 November 2014

An algebraic variety $X$ over a field $k$ into which there exists a rational mapping from a projective space, $\phi : \mathbf{P}^n \rightarrow X$, such that $\phi(\mathbf{P}^n)$ is dense in $X$ and the extension of the field of rational functions $k(\mathbf{P}^n)/k(X)$ is separable. In other words, $k(X)$ has a separable extension which is purely transcendental (cf. Transcendental extension).

Unirational varieties are close to rational varieties (cf. Rational variety), e.g. on a unirational variety there are no regular differential forms, $H^0(X,\Omega_X^p) = 0$ for $p \ge 1$. The problem of the coincidence of rational and unirational varieties is called the Lüroth problem; the answer is negative.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
How to Cite This Entry:
Unirational variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unirational_variety&oldid=34212
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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