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I've just completed exercise 2.14 in the first chapter of Hartshorne's Algebraic Geometry. This exercise asks for a proof that the image of the Segre embedding is a projective variety. The Segre embedding is the map $\mathbb{P}^r \times \mathbb{P}^s \rightarrow \mathbb{P}^{rs+r+s}, ([a_0:...:a_r], [b_0,...,b_s]) \mapsto [a_0b_0,a_0b_1,...,a_0b_s,a_1b_0,...,a_rb_s]$.

What I am wondering is: if this is to be an embedding of anything besides sets, shouldn't I at least try to prove that this is a homeomorphism onto its image (where I am thinking of the domain in the product topology)? I guess what I'm asking is what sort of imbedding is this? I.e. what structure is being preserved?

Thanks for your time!

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I think this is used to define the structure of $\mathbb{P}^r \times \mathbb{P}^s$ as a projective variety, not the other way around –  Cocopuffs Sep 18 '12 at 2:57
    
Thanks. But so all I should care about is that this is a map of sets? Do you think its a homeomorphism? Should I care whether it's a homeomorphism? –  borges Sep 18 '12 at 2:59
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Notably, the topology of $\mathbb{P}^r \times \mathbb{P}^s$ is defined so that this map becomes a homeomorphism onto its image. –  Zhen Lin Sep 18 '12 at 2:59
    
Zhen, are you just referring to the topology you can bull back from the image? I guess I'm asking whether this is the same as the product topology on the domain –  borges Sep 18 '12 at 3:04
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This map is a closed embedding into $\mathbb{P}^{rs+r+s}$ which means that the image is a closed algebraic subscheme (or rather subvariety). To show this show that in any affine patch of $\mathbb{P}^{rs+r+s}$ the points of the Segre embedding correspond to an ideal. Writing co-ordinates for $\mathbb{P}^{rs+r+s}$ as $k[X_{ij}]$ where $0\leq i\leq r$ and $0 \leq j \leq s$ we see that the image of the Segre embedding correspond to the ideal generated by all $2 \times 2$ minors of the matrix $(X_{ij})$. This shows that the image is closed. –  s.b Sep 18 '12 at 3:06

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