Figure 1

Sketch of the geometric construction. $S$ is any point in the space $\mathrm{P}{\mathbb{Q}}^{13}$ defined by the linear anomaly cancellation equations, $C$ is any point in $\mathrm{P}{\mathbb{Q}}^{13}$ satisfying the quadratic equation and $B$ is the double point of both the quadratic and the cubic equation. $L$ is the line $CS$, which generically intersects the quadratic at $R$. $M$ is the line $BR$ which lies in the quadratic and generically intersects the cubic at $X$, yielding a solution to all anomaly cancellation equations.