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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.

Igor Dolgachev studies algebraic geometry, including the classical geometry of plane curves and their symmetries. With Michela Artebani, he wrote a study of the Hesse pencil and its associated configurations.^[ref]

The pencil combines plane cubic algebraic curves sharing nine base points. Those points encode the flex geometry of a smooth cubic, while special members split into lines. The paper connects this classical configuration with elliptic surfaces and K3 surfaces, illustrating how a one-parameter family of equations organizes several geometric objects at once.^[ref]

Michela Artebani works on algebraic geometry. Her joint paper with Igor Dolgachev examines the Hesse pencil of plane cubic algebraic curves, its flex configuration and associated K3 surfaces.^[ref]

A useful feature of the example is that a smooth cubic and its Hessian belong to a tightly organized family. Incidences among the nine common points and twelve distinguished lines expose symmetries that are harder to see in an arbitrary cubic equation. Their treatment connects explicit coordinates with the global geometry of surfaces.^[ref]