It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ of smooth projective quartic hypersurfaces with a rational point (possibly after allowing for the dimension to be increased further)?
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2$\begingroup$ I am aware of no such result, but I also doubt very much that the standard approach (Morin-Predonzan, Harris-Mazur-Pandharipande, Paranjape-Srinivas) can work. That approach uses existence not just of rational points, but of rationally defined linear spaces. For instance, if you consider a general quartic $x_0^3x_1 + x_0^2(x_2^2+\dots+x_n^2)+x_0G + H$ that contains $p=[1,0,\dots,0]$, there is no line containing $p$ contained in the hypersurface that is defined over $\mathbb{R}$. So that approach seems unlikely to work. $\endgroup$– Jason StarrCommented Jul 23, 2015 at 13:36
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