= Fermat quartic surface
{c}
{title2=$x_0^4+x_1^4+x_2^4+x_3^4=0$}
Over $\mathbb C$, the four partial derivatives show that this projective surface is smooth. Adjunction makes its <canonical bundle> trivial, and the hypersurface <exact sequence> gives $H^1(\mathcal O_X)=0$, so it is a <K3 surface>. Lines given by $x_0=\zeta x_1$, $x_2=\eta x_3$, with $\zeta^4=\eta^4=-1$, lie on it. Choosing both roots differently gives skew lines, one of which is a section of the elliptic pencil defined by planes through the other.
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