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show/hide this revision's text 2 correction of the Godeaux surface

In positive characteristic, the (étale) fundamental group of a rationally connected variety is finite (Kollár, Inventiones Math., 1993).

And this happens: for Let us assume that the base field has characteristic $p$, where $p\neq 5$ and $p\not\equiv 1\pmod 5$. Then, the Godeaux surface hypersurface with equation $X_0^5+\dots+X_3^5=0$ in $\mathbf P^3$ is unirational; so is its quotient by the obvious action of $\mu_5$, \mu_5$ (a surface of general type known as the Godeaux surface), which is then therefore an unirational variety with fundamental group $\mathbf Z/5$.

However, Ekedahl has proved that this group is prime to the characteristic. I discussed that in my Bourbaki Seminar talk, « Points rationnels et groupes fondamentaux, applications de la cohomologie $p$-adique », Astérisque 294, p. 125-146.
show/hide this revision's text 1

In positive characteristic, the (étale) fundamental group of a rationally connected variety is finite (Kollár, Inventiones Math., 1993).

And this happens: for $p\neq 5$ and $p\not\equiv 1\pmod 5$, the Godeaux surface with equation $X_0^5+\dots+X_3^5=0$ in $\mathbf P^3$ is unirational; so is its quotient by the obvious action of $\mu_5$, which is then an unirational variety with fundamental group $\mathbf Z/5$.

However, Ekedahl has proved that this group is prime to the characteristic. I discussed that in my Bourbaki Seminar talk, « Points rationnels et groupes fondamentaux, applications de la cohomologie $p$-adique », Astérisque 294, p. 125-146.