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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.^[ref]

For cubic generators $F_0,F_1$, the incidence model isProjection 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.

Rick Miranda works on algebraic geometry and has written expositions of algebraic curves and elliptic surfaces. His Colorado State University page identifies both the research field and his books.^[ref]

In The Basic Theory of Elliptic Surfaces by Rick Miranda, a pencil of plane cubics gives a concrete surface construction. Blowing up its nine base points resolves the rational map to the parameter line and produces an elliptic fibration. The example connects equations for curves, blowups and the geometry of an entire surface.^[ref]