In his paper "Examples of Calabi-Yau 3-manifolds with complex multiplication", Jan Christian Rohdes claims that the surface $S \subset \mathbb{P}^3$, with variables $(y_2: y_1: x_1: x_0)$, given by the following equation is smooth
$(y_2^3-y_1^3)y_1+(x_1^3-x_0^3)x_0=0$
He says: "By using partial derivatives of the defining equation, one can easily verify that $S$ is smooth". However, the equation and its partial derivatives seem to vanish at all points with $y_1=x_0=0$. Where am I making a mistake?
Thanks for your help!
The partial derivative with respect to $y_1$ does not vanish identically when $x_0 = y_1 = 0$.
(It is given by $y_2^3 - 4y_1^3$!)