Some Taylor varieties with null Hessian

Technical Summary: Some Taylor Varieties with Null Hessian

Problem Statement
The paper addresses the classification of hypersurfaces $X = V(f) \subset \mathbb{P}^N$ with identically vanishing Hessian determinants ($h_f \equiv 0$). Classically, such hypersurfaces are characterized by algebraically dependent partial derivatives, implying that the Gauss map has a non-maximal rank and the variety is "tangentially degenerate" or "developable." While it is known that cones possess this property, the classification of non-conical hypersurfaces with vanishing Hessian in dimensions $N \geq 3$ is a classical problem initiated by P. Gordon and M. Noether. The authors focus on a specific family of algebraic varieties known as Taylor varieties ($T^n_{d,e,m}$), introduced recently by Conca, Naldi, Ottaviani, and Sturmfels [CNOS23]. These varieties arise from the Taylor expansion of rational functions in $n$ variables. Previous work established that Taylor varieties with $m = d+1$ have vanishing Hessians. The central problem of this paper is to investigate the case where $m = d+2$ (specifically for $n=2$) to determine if these varieties also yield new examples of non-defective hypersurfaces with identically null Hessian.

Methodology
The authors employ a combination of algebraic geometry and linear algebra techniques centered on the Padé matrix associated with Taylor varieties.

1. Taylor Varieties and Padé Matrices: The Taylor variety $T^n_{d,e,m}$ is defined as the projective closure of the image of a map $\psi$ sending pairs of polynomials $(P, Q)$ to the coefficients of the Taylor expansion of $P/Q$ up to order $m$. The variety is described via the vanishing of the maximal minors of a Padé matrix $P_T$. When $T^n_{d,e,m}$ is a non-defective hypersurface, its defining equation is given by $f = \det(P_T)$.
2. Column Operations and Invariance: To analyze the Hessian of $f$, the authors introduce a specific set of elementary column operations on the Padé matrix $P_T$. They replace columns in blocks $C_j$ (for $j > d-e+2$) with linear combinations of columns from a specific reference block $C_{d-e+2}$, utilizing auxiliary variables $\lambda$.
3. Differentiation and Kernel Analysis: By Lemma 4.2, these operations preserve the determinant ($f$). Since $f$ is independent of the auxiliary variables $\lambda$, the partial derivatives of $f$ with respect to $\lambda$ must vanish. This condition yields a system of linear equations involving the partial derivatives of $f$ with respect to the coefficients $c_\gamma$.
4. Hankel Matrix Construction: These linear relations are organized into a Hankel matrix $M$. The authors demonstrate that the vector of partial derivatives of $f$ lies in the kernel of $M$. By analyzing the dimensions of $M$ and the number of variables, they prove that the rank of $M$ is strictly less than the dimension of the space of partial derivatives.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The authors prove that for $n=2$, the Taylor variety $T^2_{d,e,d+2}$, provided it is a non-defective hypersurface, has an identically vanishing Hessian ($h_f \equiv 0$).
• Specific Case Analysis: The paper provides a detailed analysis of the smallest non-trivial example, $T^2_{5,4,7}$ (where $d=5, e=4, m=7$). They explicitly construct the $15 \times 15$ Padé matrix for this case and demonstrate the existence of a non-trivial kernel for the associated linear system of partial derivatives, confirming the vanishing Hessian without computing the full Hessian determinant.
• General Proof: The methodology used for the specific case is generalized in Section 4 to prove the result for arbitrary $d$ and $e$ under the non-defective hypothesis. The proof relies on showing that the image of the polar map (the gradient of $f$) is contained in a proper linear subspace defined by the vanishing of maximal minors of the matrix $M$.
• Computational Verification: The authors provide Macaulay2 code (Section 5) to construct Padé matrices and verify the non-singularity of the determinant for specific parameter choices, ensuring the varieties are indeed hypersurfaces.

Significance and Claims
The paper claims to provide new examples of non-defective hypersurfaces with identically null Hessian that are not cones. Specifically, it extends the known results of [CNOS23] (which covered the $m=d+1$ case) to the $m=d+2$ case for two variables.

The authors position their work within the historical context of the classification of null Hessian hypersurfaces, noting that while Perazzo cubics and other examples exist, finding new families is significant. The paper explicitly states that these Taylor varieties offer a "new source of varieties with vanishing Hessian." The significance lies in the algebraic mechanism used: the authors demonstrate that the specific structure of the Padé matrix for $m=d+2$ inherently forces algebraic dependence among the partial derivatives of the defining polynomial, thereby guaranteeing a degenerate Gauss map. The paper does not claim to solve the full classification problem for all dimensions or parameters but restricts its scope to proving the vanishing Hessian property for this specific family of Taylor varieties.