Extremal divisors in the Hilbert scheme of points on $\mathbb{P}^{2}$ are preserved under residuality

TL;DR

This paper proves that extremal divisors in the Hilbert scheme of points on the plane remain extremal under residuality, providing insight into the geometric structure of these schemes.

Contribution

It establishes that the extremal property of divisors is preserved under residuality in the Hilbert scheme of points on , a new result in algebraic geometry.

Findings

Extremal divisors are preserved under residuality.

The result applies to specific cases where n= or n=.

Advances understanding of the geometry of Hilbert schemes.

Abstract

Let $n=\frac{r(r+1)}{2}$ or $n=r(r+1)$. We prove that the property of being extremal is preserved under residuality on the Hilbert scheme of $n$ points in the plane.