View source for Unirational variety

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 in general, although all unirational [[algebraic curve]]s are rational.

====References====
<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>


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> János Kollár, Karen E. Smith, Alessio Corti, "Rational and Nearly Rational Varieties" , Cambridge Studies in Advanced Mathematics '''92''', Cambridge University Press (2004) ISBN 0-521-83207-1 </TD></TR>
</table>

[[Category:Algebraic geometry]]

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