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Let's consider spatial algebraic curve $C\subset \mathbb P^3$. How could I describe a family of such curves, for example the set of all curves genus $g$ passing through $k$ points?

I'd like to some "computationable" description(and then use macaulay2 ), so in case of rational curves we can parametrize them and say that my set is some linear subspace of coefficient space.

For example, how can I describe a pencil of elliptic cubics degree 4 (3,5,6 ?) passing through 4 points?

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Do you mean "algebraic curve defined by sparse equations"? –  Dima Pasechnik Apr 3 '12 at 16:58
sorry, I mean "spatial" –  Nikita Kalinin Apr 3 '12 at 21:18

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An elliptic curve of degree 4 in $P^3$ is an intersection of a pencil of quadrics. If you want it to pass through given 4 points you need all quadrics in the pencil to pass through them too. So, take the space of all qaudrics passing through the points, it has (linear) dimension $6$. Now consider the Grassmannian of all 2 dimensional subspaces there. its open part (which can be described explicitly) parameterizes the curves. So, the answer is an open subset of $Gr(2,6)$.

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for this situation, you are right, but what about other situations ? Or there is no general construction and in each situation approach have to be different? In this case, could you give me some good reference about construction of spatial algebraic curves? –  Nikita Kalinin Apr 5 '12 at 20:26

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