Interpolation in Algebraic Geometry

TL;DR

This paper explores the problem of interpolation in algebraic geometry, extending known results about rational normal curves to higher-dimensional varieties, and introduces multiple equivalent formulations and classifications of interpolation.

Contribution

It generalizes interpolation results to varieties of all dimensions, provides twenty-two equivalent formulations, and classifies when Castelnuovo curves and del Pezzo surfaces satisfy interpolation.

Findings

Smooth varieties of minimal degree satisfy interpolation.

Castelnuovo curves satisfy weak interpolation under certain conditions.

Del Pezzo surfaces satisfy weak interpolation.

Abstract

This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points and linear spaces. After some expository material regarding scrolls, we consider the generalization of this question to varieties of all dimensions and explain why smooth varieties of minimal degree satisfy interpolation. We give twenty-two equivalent formulations of interpolation. We also classify when Castelnuovo curves satisfy weak interpolation. In the appendix, we prove that del Pezzo surfaces satisfy weak interpolation. Our techniques for proving interpolation include deformation theory, degeneration and specialization, and association.