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The Segre map gives an embedding of the Segre variety $\Sigma_{n,m}$ (i.e. of the categorical product of two projective spaces of dimension $n$ and $m$) into a projective space of dimension $nm+n+m$. Is this the smallest possible dimension for any embedding into a projective space?

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    $\begingroup$ Do you mean any embedding of $\mathbf P^n \times \mathbf P^m$? If so then no; every variety of dimension $d$ embeds into $\mathbf P^{2d+1}$ (proof: projection!), and for almost all values of $m$ and $n$, the number $2(m+n)+1$ will be much smaller than $nm+n+m$. $\endgroup$
    – user64687
    Apr 11, 2014 at 15:34
  • $\begingroup$ @AsalBeagDubh Can you elaborate "proof: projection!" a little bit? Or maybe drop a name I can google to get a reference to the cited result? $\endgroup$ Apr 11, 2014 at 16:12
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    $\begingroup$ @AsalBeagDubh OK, I found a reference here: "Over an algebraically closed field, any projective smooth variety of dimension $n$ can be embedded in $\mathbb P^{2n+1}$. This is elementary and can be found in Shafarevich's Basic Algebraic Geometry, Chapter II, §5.4 ." So I now also know that "proof: projection" means that I have to find a suitable point outside of the variety, such that the projection from this point is an embedding into a space with lower dimension. Then I repeat this process until I can no longer find such points... Thanks for your answer. $\endgroup$ Apr 11, 2014 at 19:56
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    $\begingroup$ Dear Thomas, yes, sorry to be cryptic. What you say is exactly correct. The reason that $2d+1$ is the best you can do is that this is the dimension of the secant variety of (most) varieties of dimension $d$. $\endgroup$
    – user64687
    Apr 13, 2014 at 14:27

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