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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!

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Your mistake is that the partial derivative with respect to $y_1$ is $y_2^3-4y_1^3$, and does not vanish just because $y_1=x_0=0$ . Actually Rhodes is right : his surface $S$ is smooth. –  Georges Elencwajg May 15 '12 at 8:25
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This is a bit late, but let me point out that the author's family name is Rohde, with no letter 's' in sight. –  Asal Beag Dubh May 6 '13 at 19:25

2 Answers 2

The partial derivative with respect to $y_1$ does not vanish identically when $x_0 = y_1 = 0$.

(It is given by $y_2^3 - 4y_1^3$!)

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I guess you just want to check your calculations.

The derivative w.r.t. $x_0$ is $x_1^3-4x_0^3$, the one w.r.t. $x_1$ is $3x_1^2 x_0$, the one w.r.t. $y_1$ is $y_2^3-4 y_1^3$ and the one w.r.t. $y_2$ is $3y_2^2 y_1$ and they clearly have only the trivial solution in common.

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