The discriminant locus $D=\partial\mathcal{U}_{v}\cap\partial\mathcal{U}_{w}$ does not have to be affine linear. Even though $D$ lies in the polyhedral boundary $\partial\mathcal{U}_{v}$, the boundary $\partial\mathcal{U}_{w}$ will be wiggled in the affine coordinates unless the partial Legendre transform is linear.