Nash conditional independence curve

TL;DR

This paper investigates the geometric structure of the Nash conditional independence curve in binary graphical models, revealing its smoothness, irreducibility, and universality properties in algebraic geometry.

Contribution

It characterizes the Nash CI curve as a smooth irreducible complete intersection and establishes its universality in representing various algebraic varieties.

Findings

The Nash CI curve is a smooth irreducible complete intersection.

Explicit formulas for the degree and genus of the curve are provided.

The curve exhibits universality, representing diverse algebraic varieties.

Abstract

We study the Spohn conditional independence (CI) variety $C_X$ of an $n$-player game $X$ for undirected graphical models on $n$ binary random variables consisting of one edge. For a generic game, we show that $C_X$ is a smooth 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 prove two universality theorems for $C_X$: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in $\mathbb{R}^m$ defined by at most $m-1$ polynomials is isomorphic to an affine open subset of $C_X$ for some game $X$.