AI 中文总结
该研究证明所有既约实平面曲线奇点都存在实形变，通过提出迹映射方法，克服非实分支共轭对的障碍，解决了相关长期问题。
AI 中文摘要
我们证明了每一条既约实平面曲线奇点都存在实形变，这解决了A'Campo与Gusein-Zade提出的一个问题，后续该问题被Leviant-Shustin、Fomin-Pylyavskyy-Shustin-Thurston列为猜想。特别地，我们克服了非实分支共轭对带来的主要障碍。我们的核心新要素是一种构造，它从两个正规化圆盘的结点光滑化中产生分歧图，我们称之为迹映射。对于实分支，该构造重现了Gusein-Zade利用切比雪夫多项式的构造；对于具有不同切线的复共轭分支对，该构造根据普瓦松数据给出了分歧图的显式公式。该通用方法是将迹映射与A'Campo的平移和收缩相结合，为所有既约实平面曲线奇点生成分歧图和实形变。
英文摘要
We prove that every reduced real plane curve singularity admits a real morsification. This settles a question of A'Campo and Gusein-Zade, later stated as conjectures by Leviant--Shustin and by Fomin--Pylyavskyy--Shustin--Thurston. In particular we overcome the main obstruction that remained posed by conjugate pairs of nonreal branches. Our new main ingredient is a construction that produces the divide from a nodal smoothing of two normalization disks. This is what we call the trace map. For real branches, it recovers Gusein-Zade's construction using Chebyshev polynomials. For pairs of complex conjugate branches with distinct tangents, the construction gives an explicit formula for the divide in terms of the Puiseux data. The general method consists in a delicate combination of the trace map with A'Campo's translations and contractions to produce divides and real morsifications for all reduced real plane curve singularities.
Comments35 pages
