The Fermat family is given explicitly as

$$X_{s}=\{Z_{0}Z_{1}\ldots Z_{n+1}+e^{-s}\sum_{i=0}^{n+1}Z_{i}^{n+2}=0\},$$

(10)

namely we choose $a_{m}=1$ for $m$ corresponding to the monomials $Z_{0}\ldots Z_{n+1}$ and $Z_{i}^{n+2}$, and choose $\lambda$ to be the piecewise linear function with value $0$ at the origin and $-1$ at the vertices of $\Delta$.