Pencil of plane cubics

A pencil generated by cubic forms $F_0,F_1$ is the one-parameter family $s{F}_0+t{F}_1=0$ indexed by $[s:t]\in {\mathbb{CP}}^1$. A general pair has nine transverse base points.

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.

Projection to $[s:t]$ gives an elliptic fibration with connected cubic fibers. Each of the nine exceptional divisors maps isomorphically to ${\mathbb{CP}}^1$ and is therefore a holomorphic section.

Writing $H$ for the pullback of a line and $E_1,\dots ,E_9$ for the exceptional classes, a fiber has class $F=3H-{\sum }_i{E}_i$, while the blowup formula gives

If a smooth connected complex curve $C\subset E(1)$ satisfies $C^2=k$ and has degree $d=C\cdot F$ over the base, the adjunction formula gives