AI 中文总结
本文针对维数至多四的光滑真整簇，构造了沿光滑中心的有限爆破序列，以射影端点为终点，并保持所有极大拟射影开集的交集，该构造在函子性主化过程下自然且同构不变。
AI 中文摘要
设 $X$ 是特征零域上的维数至多为四的光滑真整簇。我们构造了一个沿光滑中心的有限序列的爆破，其终点是射影的，并保持 $X$ 的所有极大拟射影开集的交集。该构造仅依赖于一个固定的函子性主化过程，并且在同构下是自然的，包括在不同基域上的同构。证明分为两部分。正则局部曲面上的余长度论证给出了在每个维数下的有限预备步骤，在此之后某个极大拟射影开集的补集在 $X$ 中映射到余维数至少为三的子簇。在维数至多为四的情况下，这些像为点或曲线。该补集的归一化分量上的丰富数值类给出了线性系统基理想求和的有限界限。所得的理想下降回原域；几何整性不是必需的。
英文摘要
Let $X$ be a smooth proper integral variety of dimension at most four over a field of characteristic zero. We construct a finite sequence of blowups along smooth centers with projective endpoint, preserving the intersection of all maximal quasi-projective opens of $X$. The construction depends only on a fixed functorial principalization procedure and is natural under isomorphisms, including those over different ground fields. The proof has two parts. A colength argument on regular local surfaces gives, in every dimension, a finite preparation after which some maximal quasi-projective open has complement mapping into codimension at least three in $X$. In dimensions at most four these images are points or curves. Ample numerical classes on the normalized components of that complement give finite bounds for summing base ideals of linear systems. The resulting ideal descends to the original field; geometric integrality is not required.
Comments10 pages
