摘要翻译：
我们研究了曲线奇异点的一类特殊变形：奇异点分裂成若干个，从而保持总不变量的情况。这些也被称为等正规化或等泛型变形。我们主要限制于具有光滑分支的奇点的变形。引入了奇异型自然不变量：对偶图。它对可能的碰撞/变形施加了严格的限制。并证明了经典不变量在等正规化族中的变分的一些界。我们详细讨论了普通多点的变形，奇点在普通多点集合中的变形，以及$x^p+y^{pk}$在$a_k$集合中的变形。
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英文标题：
《On the equi-normalizable deformations of singularities of complex plane
  curves》
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作者：
Dmitry Kerner
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches.   A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families.   We consider in details deformations of ordinary multiple points, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type $x^p+y^{pk}$ into the collections of $A_k$'s.
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PDF链接：
https://arxiv.org/pdf/0805.4083