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

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Products of nonempty projective varieties exist as projective varieties

Statement

Nonempty projective varieties $X \subseteq P_{k}^{m}$ and $Y \subseteq P_{k}^{n}$ have a product, realized as their Segre image, and that product is a projective variety.

Facts & Assumptions

Given: Nonempty projective varieties $X \subseteq P^{m}$ and $Y \subseteq P^{n}$ .

Proof

1.1

Inside the rank-one locus, the opens $z_{i_{0} j_{0}} \neq 0$ cover. On each such open, the regular inverse coordinates from the Segre theorem identify the desired subset with the product of the corresponding affine pieces of $X$ and $Y$ , hence with a closed subset of that chart. Closedness is local on this finite open cover, so the restricted Segre image is a closed projective algebraic set.

given

2.1

The inverse coordinate recovery in the Segre theorem identifies this set with pairs $(x, y) \in X \times_{s e t} Y$ ; the displayed regular ratios show on every chart that both coordinate projections are morphisms. Thus the Segre point map and its inverse are morphisms for this constructed structure.

step 1.1

3.1

A pair of morphisms into $X, Y$ has on each inverse-image product chart the Segre coordinate formula $[f_{i} g_{j}]$ . These local formulas are regular and agree on overlaps, so they give a morphism into the closed model. The coordinate projections recover the given maps, and the point-pair identification makes the factorization unique. Hence this model has the product universal property.

step 2.1

4.1

The nonempty standard affine opens of $X$ and $Y$ are affine varieties. Their pairwise products are affine varieties by the affine-product theorem, and they cover the Segre model. Any two such product opens meet: their factor opens meet by irreducibility of $X$ and $Y$ , and choosing one point from each of those two nonempty intersections gives a point of both product opens. Thus the covering affine opens all meet, so their union is irreducible. The model is therefore a nonempty projective variety.

step 3.1∎

Depends on

Used by

The Segre-Veronese map is a closed embedding Corollary
Base change of classical varieties when the pullback exists Definition
Incidence correspondence loci Definition
The Segre image of P1 times P1 is a quadric surface Example
Fixed-multidegree forms define maps from products to projective space Theorem
Graphs of morphisms to a projective classical variety are closed Theorem

Dependency tree · two levels

9 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, §5g and 6.26 (standard reference, not scraped)