无非退化奇异方向的单位芽切线的双有理性质
Birational properties of tangent to the identity germs without non-degenerate singular directions
摘要：
我们提供了一个家庭的孤立切线的身份芽$f:(\mathbb{C}^3,0) \to (\mathbb{C}^3,0)$只拥有退化的特征方向，并为其中的提升$f$的任何修改（与适当的属性）只有退化的特征方向。这与维度$2$中的情况形成鲜明对比，在维度$2$中，任何与恒等胚$f$相切的孤立切线都允许修改，其中$f$的提升具有非退化的特征方向。我们比较这种情况与奇点的分解的无穷小发电机的$f$，表明这种现象是不相关的向量场的Gomez-蒙特和Luengo的复分离线的不存在。最后，我们描述了一组正式的$f$不变曲线，和相关的抛物流形，使用最近开发的技术由López-Hernanz，Raissy，Ribón，Sanz Sánchez，Vivas。
英文摘要：
We provide a family of isolated tangent to the identity germs $f:(\mathbb{C}^3,0) \to (\mathbb{C}^3,0)$ which possess only degenerate characteristic directions, and for which the lift of $f$ to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension $2$, where any isolated tangent to the identity germ $f$ admits a modification where the lift of $f$ has a non-degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of $f$, showing that this phenomenon is not related to the non-existence of complex separatrices for vector fields of Gomez-Mont and Luengo. Finally, we describe the set of formal $f$-invariant curves, and the associated parabolic manifolds, using the techniques recently developed by López-Hernanz, Raissy, Ribón, Sanz Sánchez, Vivas.
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