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]]). | 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]]). | ||
| − | 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. | + | 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 in general, although all unirational [[algebraic curve]]s are rational for algebraically closed $k$. |
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Revision as of 12:08, 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 in general, although all unirational algebraic curves are rational for algebraically closed $k$.
References
| [1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Unirational variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unirational_variety&oldid=34213