A moduli scheme parametrizing blowups of smooth projective surfaces

TL;DR

This paper constructs and analyzes a moduli scheme parametrizing sequences of blowups of smooth projective surfaces, proving its smoothness, projectivity, and describing its Chow ring structure.

Contribution

It introduces a new moduli scheme for blowups of surfaces, proves its geometric properties, and explicitly describes its Chow ring, especially over complex rational surfaces.

Findings

The moduli scheme $F[n]$ is smooth and projective.

Divisors $D_{i,j}^{(n)}$ correspond to point identifications under projections.

The Chow ring $ ext{A}^*(F[n])$ is generated by these divisors over $ ext{A}^*(S_1^n)$.

Abstract

We construct a moduli scheme $F[n]$ that parametrizes tuples $(S_1, S_2, \dots, S_{n+1}, p_1, p_2, \dots, p_n)$ in which $S_1$ is a fixed smooth surface over $\text{Spec } R$ and $S_{i+1}$ is the blowup of $S_i$ at a point $p_i$, $\forall 1\leq i\leq n$. We show that this moduli scheme is smooth and projective. We prove that $F[n]$ has smooth divisors $D_{i,j}^{(n)}$, $\forall 1\leq i<j\leq n$, which correspond to tuples that map $p_j\mapsto p_i$ under the projection morphism $S_j\to S_i$. When $R=k$ is an algebraically closed field, we demonstrate that the Chow ring $\mathbb{A}^*(F[n])$ is generated by these divisors over $\mathbb{A}^*(S_1^n)$. We end by giving a precise description of $\mathbb{A}^*(F[n])$ when $S_1$ is a complex rational surface.