The Unirationality of Quartics over Nonclosed Fields Revisited

N. F. Zak

Central Bank of the Russian Federation

Abstract: Examples of smooth fourth-degree hypersurfaces which are unirational over an algebraically nonclosed field $\Bbbk$ and contain no straight lines defined over $Bbbk$ are given. A method for proving the unirationality of quartics is suggested, which, unlike other methods, does not use linear spaces contained in the quartics.

Keywords: unirational quartic, quartic over an algebraically nonclosed field, unirational variety, irreducible hypersurface, Plucker embedding, birational projection.

UDC: 512.626

Received: 18.07.2007
Revised: 03.02.2008

DOI: 10.4213/mzm4030

 English version: Mathematical Notes, 2008, 84:1, 38–44

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