= Rational elliptic surface
{title2=$E(1)$}
{wiki}
A rational elliptic surface is a <rational surface> carrying an <elliptic fibration>; in the standard section-bearing case its general fibre is a smooth <genus>-one curve, it has a section, and no smooth rational curve of self-intersection $-1$ is contained in a fibre. A general <pencil of plane cubics> gives the model $E(1)$ by blowing up its nine transverse base <points>.<The Basic Theory of Elliptic Surfaces by Rick Miranda>[Example (I.5.1), printed p.7]{ref}
For cubic generators $F_0,F_1$, the incidence model is
$$
E(1)=\{([z],[s:t])\in\mathbb{CP}^2\times\mathbb{CP}^1:
sF_0(z)+tF_1(z)=0\}.
$$
Projection to the <projective plane> identifies this surface with its nine-point <blowup>. Projection to the <projective line> gives the <elliptic fibration of the rational elliptic surface>. Over each base <point> the incidence fibre is one projective line, the corresponding <exceptional divisor>, and each such divisor is a section. The surface is rational because the blowup is birational to the <projective plane>.
The ambient space in this equation is the original projective plane times the parameter line. If one first pulls $F_0,F_1$ back to the blowup, both have common exceptional factors. Their unmodified incidence equation in the blowup times the parameter line would then incorrectly include an entire exceptional-divisor-times-line over each base <point>; those common factors must be removed to describe the graph there.
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