How many curves of degree n pass through a set of d point?

Question

I'm really new to projective geometry and I need help with that question. Given a set of d fixed point how many curves of degree n in $\mathbb{P}^2$ passes through these points and why? As I said, I'm really new to the subject and I don't even know if the question is specific enough. I am quite sure this has to do something with the Hilbert function, but I don't know how to compute it and how to use it.

Answer 1

If by "curve of degree $n$" you mean the zero-set of an $n$-th degree polynomial in two variables, such polynomials have $\binom{n+2}2$ coefficients, one of which can always be set to $1$ by scaling. Each point specified to be on the curve provides an equation. With $\binom{n+2}2 - 1$ points we have $\binom{n+2}2 - 1$ equations in the $\binom{n+2}2 - 1$ unknown coefficients. Note that all of these are linear equations in the coefficients.

So,

• with fewer than $\binom{n+2}2 - 1$ points, the coefficents are underspecfied, and there are infinitely many curves.
• with exactly $\binom{n+2}2 - 1$ points, in general, there will be only one curve passing through those points. However, it is possible for the points to be aligned just right for the system of linear equations in the coefficients to be singular. In this case (which has probability $0$ if one chooses points at random from some specified bounded domain), infinitely many curves will pass through those points.
• with more than $\binom{n+2}2 - 1$ points, the system is overspecified, and in general will have no solutions and thus no curves passing through all points. Again, it is possible (with probability $0$) for the points to be positioned just right to allow either one curve, or infinitely many curves pass through.

While I described this in terms of finite points, this still holds true for points at $\infty$. Each of them expresses a limiting condition on the polynomial which also amounts to a linear equation in the coefficients.