On the geometry of the Hilbert schemes of points in the projective plane

TL;DR

This paper studies the geometric structure of Hilbert schemes of points on the projective plane, classifying minimal-degree curves and determining key geometric features like the nef cone and flip structures.

Contribution

It provides a classification of minimal-degree curves and characterizes the nef cone and flip structure for Hilbert schemes on the projective plane.

Findings

Classification of minimal-degree curves in Hilbert schemes.

Determination of the nef cone for these schemes.

Identification of flip structures in the geometry.

Abstract

We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.