Interpolation

Summary

This is an overview of interpolation problems: when, and how, do zero-dimensional schemes in projective space fail to impose independent conditions on hypersurfaces?

1. The interpolation problem

We give an overview of the exciting class of problems in algebraic geometry known as interpolation problems: basically, when points (or more generally zero-dimensional schemes) in projective space may fail to impose independent conditions on polynomials of a given degree, and by how much.

We work over an arbitrary field K. Our starting point is this elementary theorem:

Theorem 1.1. Given any, there is a unique of degree at most d such that

More generally:

Theorem 1.2. Given any, natural numbers with, and

there is a unique of degree at most d such that

The problem we'll address here is simple: What can we say along the same lines for polynomials in several variables?

First, introduce some language/notation. The “starting point” statement Theorem 1.1 says that the evaluation map

is surjective; or, equivalently,

for any distinct points whenever. More generally, Theorem 1.2 says that

when. To generalize this, let be an subscheme of dimension 0 and degree n. We say that F imposes independent conditions on hypersurfaces of degree d if the evaluation map

is surjective, that is, if

we'll say it imposes maximal conditions if p has maximal rank—that is, is either injective or surjective, or equivalently if.