CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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A degree-d homogeneous equation becomes a hyperplane section under Veronese

Statement

If $n \geq 1$ and $F$ is a nonzero homogeneous polynomial of degree $d \geq 1$ on $P^{n}$ , then there is a hyperplane $H \subseteq P^{N}$ such that $V_{+} (F) = ν_{n, d}^{- 1} (H)$ .

Proof

Given: $F = \sum_{α} c_{α} x^{α}$ of degree $d$ .

1.1

In the ordered Veronese coordinates $Z_{α}$ , define the linear form $L = \sum_{α} c_{α} Z_{α}$ and its hyperplane $H = V_{+} (L)$ .

given

2.1

Substitution in the definition of $ν_{n, d}$ gives $L (ν_{n, d} ([x])) = \sum c_{α} x^{α} = F (x)$ .

step 1.1algebra

3.1

Therefore a point belongs to the pullback hyperplane exactly when it belongs to $V_{+} (F)$ , proving the equality.

step 2.1∎

Depends on

Used by

The quadratic Veronese image of P1 is a plane conic Example

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, 6.24 (standard reference, not scraped)