A moduli scheme parametrizing blowups of smooth projective surfaces

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 $p_i$, $\forall 1\le i\le n$. We show that this moduli scheme is smooth and projective. We prove that $F[n]$ has smooth divisors $D_{i,j}^{(n)}$, $\forall 1\le i<j\le n$, 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 ${\mathbb{A}}^{\ast }(F[n])$ is generated by these divisors over ${\mathbb{A}}^{\ast }(S_1^n)$. We end by giving a precise description of ${\mathbb{A}}^{\ast }(F[n])$ when $S_1$ is a complex rational surface.