The Euler Stratification for ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^n$

Abstract

We study the Euler characteristic of a hypersurface in $({\mathbb{C}}^{\ast }{)}^2\times ({\mathbb{C}}^{\ast }{)}^n$ defined by a polynomial whose monomial support corresponds to lattice points in $\hat{I}{”}_1\times \hat{I}{”}_1\times \hat{I}{”}_n$ as the coefficients of the defining polynomial vary. Each member of this hypersurface family corresponds to a three-way independence model from algebraic statistics, and the (signed) Euler characteristic is equal to the maximum likelihood degree (ML degree) of the model. We show in the case of $\hat{I}{”}_1\times \hat{I}{”}_1\times \hat{I}{”}_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $\hat{I}{”}_1\times \hat{I}{”}_1\times \hat{I}{”}_n$ with $n\ge 2$. We prove that, for all $n\ge 1$, all positive integers up to the maximum possible ML degree can be realized as the Euler characteristic. Furthermore, we completely determine the Euler stratification for ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ and provide partial information for ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^2$.

31 pages, 7 figures, minor revisions to section 4, added new figure