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 $(\mathbb{C}^*)^2 \times (\mathbb{C}^*)^n$ defined by a polynomial whose monomial support corresponds to lattice points in $Δ_1 \times Δ_1 \times Δ_n$ 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 $Δ_1 \times Δ_1 \times Δ_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $Δ_1 \times Δ_1 \times Δ_n$ with $n \geq 2$. We prove that, for all $n\geq 1$, 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 $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$.

Figures (6)

• Figure 1: Two different $2 \times 2$ determinant factors and one $2 \times 2 \times 2$ hyperdeterminant factor of $E_A$ for $n = 2$
• Figure 2: Vanishing relations among factors of $E_A$ for the case $n = 1$
• Figure 3: The red, blue, purple, and green quadrics are the vanishing loci of $f_0$, $f_1$, $f_2$, and $f_2'$, respectively.
• Figure 4: Two nonsingular quadrics $Q_j, Q_k$ where $F_{\bullet 1 (j,k)} = F_{0 \bullet (j,k)} = 0$
• Figure 5: The alternating hook minors $\mathrm{altH}(4)$

Theorems & Definitions (87)

• Proposition 1.1
• Proposition 1.2
• Conjecture 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Definition 2.2