There's an algebraic geometry theorem (I.4.9 in Hartshorne) that says: any variety of dimension r (over an algebraically closed field) is birationally equivalent to a hypersurface in projective space of dimension r+1. The proof is quite algebraic, and I'd like to see some interesting geometric examples.
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The proof is actually extremley geometric, if you just wave your hand hard enough. Take an r-dimension variety V in P^{r+d}. Pick a generic point Q in P^{r+d} (where d > 1), and project P^{r+d} from this point to P^{r+d-1}. A generic line through Q does not hit V at all, and a generic line that hits V, hits it at exactly one point (I waved my hands right here), so the projection is birational from V to it's image. |
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One simple example is like this. Take the singular quadric x^2+y^2+z^2=0 in CP^3. It has 1 double point (0:0:0:1). If you resolve this point you get a CP^1 bundle over CP^1, a Hirzebrough surface. It is not hard to see that this surface can not be realised as a smooth surface in CP^3. This example can be generalised in any way you wish, just take a singular hypersuface in CP^n and blow it up. |
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