AI 中文总结
本文推广了幂级数族中完备纤维维数的半连续性结果，并应用于证明超曲面和完全交族中米尔诺数等奇点不变量的半连续性。
AI 中文摘要
本文是文献[GP21]的推广，并给出了新的应用。原问题源于正特征奇点分类。在形变给定奇点时，某些不变量不能增加，这一点很重要。在[GP21]中，我们证明了在带截面的形变下，完备纤维维数满足此性质。一般情况下，邻近纤维包含多个奇点，本文证明完备纤维维数之和也具有半连续性。对于解析族或代数族，这是众所周知的。但对于幂级数族，问题更困难，因为定义不变量的模是拟有限的，但在基空间上不是有限的。事实上，通常的纤维维数一般不具有半连续性。然而，若在任意诺特环上的环族或模族中考虑完备纤维，我们可以证明其维数具有半连续性。证明方法与[GP21]不同，虽然更一般，但甚至更简单。最后，我们应用此结果证明若干奇点不变量的半连续性，如超曲面族和完全交族中的米尔诺数和丘里纳数。我们以一个关于孤立完全交奇点的米尔诺数的问题作为结尾。
英文摘要
The present article is a generalization of our paper \cite{GP21}, together with new applications. The original problem came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants cannot increase if we deform a given singularity. In \cite{GP21} we proved this for the completed fiber dimension under deformations with section. In general the nearby fiber contains however several singularities and in this article we prove that also the sum of the completed fiber dimensions behaves semicontinuous. This is well known for analytic or algebraic families. However, for families of power series the problem is more difficult, since the the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the usual fibre dimension is not semicontinuous. However, if we pass to the completed fibres in a family of rings or modules over arbitrary Noetherian rings, we can prove that their dimension is semicontinuous. The proof is different from that given in \cite{GP21} and, though more general, even easier. Finally we apply this to prove the semicontinuity of several singularity invariants, such as the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections. We end with a problem concerning the Milnor number of an isolated complete intersection singularity.
Comments41 pages
