The conjectural structure of the hypergeometric discriminant for type B_n

Let $Z$ be the complex vector space with coordinates $m_0,\dots,m_n$, and let $V(f)\subset Z$ denote the vanishing locus of a polynomial $f$. Let $T^*_W Z$ denote the conormal variety to a subvariety $W\subset Z$, and let $\mathbb{P}(Z)=(Z\setminus\{0\})//\mathbb{C}^*$ be the projectivization of $Z$. Write $\Delta_n=\Delta_n(m_0,\dots,m_n)$ for the hypergeometric discriminant, and let $\mu_0,\dots,\mu_n$ be the dual coordinates on $\mathbb{P}(Z)^*$.

The conjectural structure of the hypergeometric discriminant for type $B_n$. The hypergeometric discriminant of $\texttt{B}_n$ is given by the formal sum

$$E^{\rm hyp}_{\texttt{B}_n}=n_0[T^*_Z Z]+[T^*_{V(\Delta_n)}Z]+\sum_{p=1}^{n-1}n_p\sum_{0\leq i_1<\cdots<i_p\leq n}[T^*_{V(m_{i_1},\dots,m_{i_p})}Z],$$

where $n_0=2^n-1$ and $n_p$ are natural numbers. The variety $V_{\mathbb{P}(Z)}(\Delta_n)$ is the projective dual to the reciprocal hyperplane

$$\frac{1}{\mu_0}+\frac{1}{\mu_1}+\cdots+\frac{1}{\mu_n}=0.$$

In particular, $V_{\mathbb{P}(Z)}(\Delta_n)$ is a rational hypersurface, with rational parametrization

$$\mathbb{P}^{n-1}\ni [t_1:\dots:t_n]\mapsto \left[\left(\sum_{i=1}^{n}t_i\right)^2:t_1^2:\dots:t_n^2\right]\in\mathbb{P}(Z).$$

The formula extends the explicitly computed type $B_4$ case and predicts that the conormal-cycle coefficients are determined by natural numbers $n_p$, with the generic coefficient $n_0=2^n-1$. The duality and parametrization describe the geometry of the principal discriminant component; the conjecture remains unproved in general.