Talk:Segre embedding

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[edit] Categorical product

FWIW, Joe Harris (in the cited ref) claims that the Segre embedding is the categorical product for projective varieties; the book however, brushes only very lightly on category theory. linas 14:19, 18 December 2006 (UTC)

The categorical product of P^m and Pⁿ is P^m×Pⁿ (defined as a fiber product if you want) in any reasonable category of algebraic varieties over a fixed field. The categorical point about Segre is that it gives the easy proof that the subcategory of projective varieties contains P^m×Pⁿ. If you like, the issue is not what the product is, but where it is. The Segre embedding is strictly a morphism, not an object. Charles Matthews 16:05, 18 December 2006 (UTC)