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Over $\mathbb{C}$, the Hesse pencil is the pencil of plane cubics generated by $x^3+y^3+z^3$ and $xyz$. In the affine parameter used here,The parameter at infinity gives the triangle $xyz=0$. Its finite member is nonsingular exactly when ${\lambda }^3\ne -1$.

Indeed, a singular point satisfies $x^2=-\lambda yz$, $y^2=-\lambda xz$, $z^2=-\lambda xy$. No coordinate can vanish, since that would make all three zero. Multiplying gives ${\lambda }^3=-1$. Conversely, these three parameter values give products of three lines; for example $\lambda =-1$ uses the factorization of $x^3+y^3+z^3-3xyz$. The Hessian curve of a finite member has equationobtained by taking the determinant of the second-derivative matrix and discarding a nonzero scalar. It remains in the pencil, explaining the common flex configuration.^[ref]

The ground field matters. In characteristic three, this displayed family collapses to $(x+y+z{)}^3=0$, so the complex smoothness test is inapplicable. The ordinary Hessian determinant also degenerates in characteristic two. The formulas above and the accompanying flex configuration of the Hesse pencil are used here over the complex numbers.

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.