Determine if the determinant of the Hessian matrix of a smooth homogeneous polynomial maintains specific algebraic properties across different variable subspaces. | Step-by-Step Solution

1. What We're Solving:

We're investigating the determinant of the Hessian matrix of a smooth homogeneous polynomial F of degree d ≥ 2 in n variables. Specifically, we want to understand how this determinant behaves when F defines a smooth hypersurface in projective space, and what algebraic properties it maintains.

2. The Approach:

We're studying how the "curvature information" (encoded in the Hessian) of our polynomial surface behaves. The key insight is that homogeneous polynomials have special scaling properties, and when they define smooth projective varieties, their Hessian matrices inherit fascinating structural properties.

3. Step-by-Step Solution:

Step 1: Set up the Hessian matrix

• For F(x₁, x₂, ..., xₙ), the Hessian H is the n×n matrix where H_{ij} = ∂²F/∂x_i∂x_j
• Since F is homogeneous of degree d, each second partial derivative has degree (d-2)

Step 2: Use Euler's theorem for homogeneous functions

• For homogeneous F of degree d: Σᵢ xᵢ(∂F/∂xᵢ) = dF
• Taking another derivative: Σᵢ xᵢ(∂²F/∂xᵢ∂xⱼ) = (d-1)(∂F/∂xⱼ)
• This means: H·x = (d-1)∇F, where x = (x₁,...,xₙ)ᵀ

Step 3: Analyze the rank of H

• The equation H·x = (d-1)∇F tells us that x is in the column space of H
• If F defines a smooth hypersurface, then ∇F ≠ 0 at smooth points
• This means H has rank ≥ 1, but the vector x is always in the kernel of H when restricted to the level set F = 0

Step 4: Investigate the determinant

• Since H·x = (d-1)∇F and we're on a hypersurface where this relationship holds, H cannot have full rank when restricted to the hypersurface
• The determinant det(H) is a homogeneous polynomial of degree n(d-2)
• On the smooth hypersurface F = 0, we expect det(H) to have special vanishing properties

Step 5: Connect to projective geometry

• In projective space, we work with equivalence classes [x₁:...:xₙ]
• The condition H·x = (d-1)∇F is preserved under scaling
• The determinant det(H) transforms predictably under coordinate changes

4. The Key Results:

For a smooth homogeneous polynomial F of degree d defining a smooth hypersurface in projective space:

• 1. Rank Property: The Hessian H has rank exactly (n-1) at smooth points of the hypersurface F = 0
• 2. Determinant Vanishing: det(H) = 0 identically on the hypersurface F = 0
• 3. Structural Property: The kernel of H at any point on F = 0 is spanned by the position vector x
• 4. Degree Property: det(H) is homogeneous of degree n(d-2)

The beautiful conclusion is that the determinant of the Hessian provides geometric information about the singularities and curvature properties of the projective hypersurface!

5. Memory Tip:

Remember "HEK" - Hessian Eigenvalues reveal Kernels! For smooth projective hypersurfaces, the Hessian always has the position vector in its kernel, which forces the determinant to vanish on the hypersurface itself. This connects the algebraic structure (vanishing determinant) to the geometric reality (smooth hypersurface).