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

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A nonempty projective cone raises dimension by one

Statement

If $X \subseteq P_{k}^{N}$ is a nonempty projective algebraic set, then $\dim C (X) = \dim X + 1$ . Over $X_{i} = X \cap D_{+} (T_{i})$ , the locus $C (X) \cap D (T_{i})$ is isomorphic to $X_{i} \times G_{m}$ . If $X$ is irreducible, so is $C (X)$ .

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Products of nonempty classical varieties exist in the category of classical varieties, and $\dim (X \times_{k} Y) = \dim X + \dim Y$ . If both factors are irreducible, their product is irreducible. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimensions add under products).

[F2]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F3]

If $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ . Every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

[F4]

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ . (Dimension of a finite closed union).

[F5]

For $X \subseteq P_{k}^{n}$ , define its affine cone $C (X) = V (I_{+} (X)) \subseteq A_{k}^{n + 1}$ . It is stable under scalar multiplication. If $X \neq \emptyset$ , then $0 \in C (X)$ ; under the stated definition, $C (\emptyset) = \emptyset$ . (affine cone projective set).

[F6]

The affine cone over a classical projective variety is irreducible. (projective variety cone irreducible).

Proof

1.1

For a nonzero cone point with $i$ th coordinate $λ \neq 0$ , normalize by dividing its coordinates by $λ$ . This gives $([v], λ) \in X_{i} \times G_{m}$ , whose inverse is multiplication of the representative with $i$ th coordinate one by $λ$ . These are regular inverse maps on the indicated affine charts. The cone is the one defined by the homogeneous vanishing ideal; the assertion assumes $X \neq \emptyset$ .

2.1

If $X$ is irreducible, its cone is irreducible by the cone supplier. Each nonempty cone chart therefore has the dimension of the whole cone. The group $G_{m} = D (t) \subset A^{1}$ is a nonempty open of dimension one, and $X_{i}$ is a nonempty open of $X$ . Product dimension gives $\dim C (X) = \dim (X_{i} \times G_{m}) = \dim X + 1$ .

F1F2F3F6step 1.1

3.1

For reducible $X$ , take its finitely many irreducible components $X_{j}$ . The cone is the finite closed union of their cones: every nonzero vector projects to some component, and the common vertex belongs to all their cones. Taking the maximum of their dimensions gives $\dim C (X) = \max_{j} (\dim X_{j} + 1) = \dim X + 1$ . In particular a point has a line as its cone.

F4step 2.1∎

Depends on

Used by

Several homogeneous equations in projective space Corollary
Nontrivial projective hypersurface sections Lemma
Projective intersection dimension and nonemptiness Theorem

Dependency tree · two levels

19 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

Milne §6p proof of Theorem 6.43 and Corollary 6.47 (standard reference, not scraped)