Complete intersections on Veronese surfaces

TL;DR

This paper classifies all reduced complete intersection point sets on Veronese surfaces, proposes a conjecture for higher dimensions, and proves it for quadratic Veronese threefolds, using Hilbert function characterizations.

Contribution

It provides a complete classification for points on Veronese surfaces and introduces a conjecture for higher-dimensional cases, proven in specific instances.

Findings

All possible reduced complete intersection point sets on Veronese surfaces are described.

A conjecture for complete intersection subvarieties of any dimension is formulated.

The conjecture is proven for the quadratic Veronese threefold.

Abstract

In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces.