Categorical blow-up formula for Hilbert schemes of points

TL;DR

This paper establishes a semi-orthogonal decomposition of the derived category of Hilbert schemes on a blow-up surface, extending known formulas for Euler characteristics using advanced categorical and geometric techniques.

Contribution

It introduces a categorical blow-up formula for Hilbert schemes of points, connecting derived categories of the blow-up and original surface.

Findings

Derived category decomposition for blow-up Hilbert schemes

Recovery of Euler characteristic blow-up formula

Application of Quot formula in categorical context

Abstract

Let $S$ be a smooth projective surface, and $\hat{S}$ be its blow-up at a point. In this paper, we study the derived category of the Hilbert scheme of points on the blow-up $\hat{S}$. We obtain a semi-orthogonal decomposition consisting of the derived categories of the Hilbert schemes on the original surface $S$, which recovers the blow-up formula for the Euler characteristics obtained by G\"ottsche and Nakajima-Yoshioka. The proof uses the Quot formula, which was conjectured by Jiang and recently proved by Toda.