Nash conditional independence curve

We study the Spohn conditional independence (CI) varieties of an $n$-player game for one-edge Bayesian networks on $n$ binary random variables. For generic payoff tables we prove that the Spohn CI variety is an irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety $(\mathbb P^{1})^{n-2} \times \mathbb P^3$ and we give an explicit formula for its degree and genus. We show that any affine real algebraic variety $S \subseteq \mathbb{R}^m$ defined by $k$ polynomials with $k