Polynomial interpolation is an important subject in many areas of mathematics, from approximating functions in analysis and curve fitting in statistics to Hilbert functions in commutative algebra. The simplest case is the problem solved by Lagrangian interpolation, which discusses when a single variable polynomial can have prescribed values at fixed points. Moving on, when the polynomials have several variables, the interpolation problem becomes much more challenging and depends heavily on the position of the points. For instance, if 3 points are collinear, then if we know the values of a linear polynomial at two of the points then we also know it at the third.

After introducing several basic interpolation problems, we will turn to studying the following problem. Given a collection of $n$ points in the plane, what is the most interesting codimension 1 property that those points can satisfy? For example, it is one condition on the coordinates of 3 points for them to lie on a line. We will phrase this problem in terms of the computation of effective divisors on the Hilbert scheme of points in the plane, and see how a natural generalization of the polynomial interpolation problem provides the answer.