论文标题
平滑射击表面的模量方案参数化攻击
A moduli scheme parametrizing blowups of smooth projective surfaces
论文作者
论文摘要
我们构造一个模块方案$ f [n] $，该$ f [n] $，该元组$（s_1，s_2，\ dots，s_ {n+1}，p_1，p_1，p_2，\ dots，p_n）$，其中$ s_1 $是$ \ text {spec} r $ and $ s_ po_ po_ a po_ a po_ a po_ a $ s $ s $ s $ s $ s $ s $ s $ s $ s $ $ \ forall 1 \ leq i \ leq n $。我们表明，这种模量方案是平稳且投射的。我们证明$ f [n] $具有光滑的除数$ d_ {i，j}^{（n）} $，$ \ forall 1 \ leq i <j \ leq n $，它与映射$ p_j \ mapsto p_i p_i $ p_i $相对应。当$ r = k $是一个代数关闭的字段时，我们证明了这些分隔线在$ \ mathbb {a}^*（s_1^n）$上生成的Chow Ring $ \ mathbb {a}^*（f [n]）$。最终，当$ s_1 $是一个复杂的理性表面时，我们将提供$ \ mathbb {a}^*（f [n]）$的精确描述。
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.
