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

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The Veronese map is a well-defined closed immersion

Statement

For $d \geq 1$ , $ν_{n, d}$ is a well-defined closed immersion of projective varieties.

Proof

Given: $d \geq 1$ and homogeneous coordinates $[x_{0} : \dots : x_{n}]$ .

1.1

Scaling $x$ by $λ$ scales every degree- $d$ monomial by $λ^{d}$ ; not all monomials vanish because some $x_{i} \neq 0$ . Thus the homogeneous-coordinate criterion makes $ν_{n, d}$ a morphism.

givenalgebra

2.1

On the open where $x_{i}^{d} \neq 0$ , the ratios $x_{j} x_{i}^{d - 1} / x_{i}^{d} = x_{j} / x_{i}$ recover the usual affine coordinates. These chart inverses show injectivity and regular local inverse maps.

step 1.1algebra

3.1

The relations $Z_{α} Z_{β} - Z_{γ} Z_{δ}$ whenever $α + β = γ + δ$ vanish on monomial coordinates; on each $Z_{d e_{i}} \neq 0$ chart they express every coordinate as a monomial in the recovered ratios. Hence they cut out exactly the image, which is closed, and step 2.1 makes the map a closed immersion.

step 2.1algebra∎

Depends on

Used by

A degree-d homogeneous equation becomes a hyperplane section under Veronese Corollary
The Segre-Veronese map is a closed embedding Corollary
The quadratic Veronese image of P1 is a plane conic Example

Dependency tree · two levels

10 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, 6.23 (standard reference, not scraped)