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I want to do something very concrete: write down a smooth scheme of given degree and dimension in projective n-space.

A natural way to go about this is to try and write down a complete intersection, but not all degrees/dimensions can be gotten this way. For instance, I want to write down a smooth, non-degenerate cubic surface in $\mathbb{P}^4$.

What's a systematic way to go about this kind of problem?

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    Why do you want intersections only when they are complete?2014-10-28
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    @ColinMcLarty: I guess all the OP was saying was that it's trivial to read off the degree and dimension of a complete intersection, whereas for non-complete intersections it can be extremely difficult.2014-10-28
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    Asal expresses my point perfectly.2014-10-28

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