I am considering the quintic:
$$ x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 - 5 x_1x_2x_3x_4x_5=0$$
and it is said (Candelas et al.) that
only one singular point which we may as well take to be the point $(1, 1, 1, 1, 1)$. The singularity is a node, that is a point for which $p$ and $dp$ both vanish but the matrix of second derivatives is nonsingular.
But I obtain that the previous statement is false, since the determinant of the Hessian is
$$ -12500 (-255 x_1^3 x_2^3 x_3^3 x_4^3 x_5^3 + 3 x_1^2 x_2^2 x_3^2 x_4^2 x_5^2 (x_1^5 + x_2^5 + x_3^5 + x_4^5 + x_5^5) + 8 x_1 x_2 x_3 x_4 x_5 (x_3^5 x_4^5 + x_3^5 x_5^5 + x_4^5 x_5^5 + x_2^5 (x_3^5 + x_4^5 + x_5^5) + x_1^5 (x_2^5 + x_3^5 + x_4^5 + x_5^5)) + 16 (x_3^5 x_4^5 x_5^5 + x_2^5 (x_4^5 x_5^5 + x_3^5 (x_4^5 + x_5^5)) + x_1^5 (x_4^5 x_5^5 + x_3^5 (x_4^5 + x_5^5) + x_2^5 (x_3^5 + x_4^5 + x_5^5))))$$
which vanishes at the point $(1,1,1,1,1)$.
This does not seem an erratum since it is mentioned in other places. The only solution I can imagine is that we are in fact considering a parameter $\psi$ as a point of the singularity, such that
$$ x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 - 5 \psi x_1x_2x_3x_4x_5=0$$
and we evaluate it at $\psi=1$, in that case, the Hessian (form deriving wrt. $\psi$ too) is non-singular.