On the topological classification of complex plane curve singularities

Abstract

This final degree project is devoted to study the topological classification of complex plane curves. These are subsets of $\mathbb{C}^2$ that can be described by an equation $f(x,y)=0$. Loosely speaking, curves are said to be equivalent in a topological sense whenever they are ambient homeomorphic, i.e., there exists an orientation-preserving homeomorphism of the ambient space carrying one curve to the other. The project's aim is to develop operative and clear conditions to determine whether two curves are equivalent. Curves will be shown to be decomposable into branches: sets that can be explicitly parametrised in the form $x=φ(t), y=ψ(t)$. These parametric expressions will be analysed to extract a complete numerical invariant for the classification of branches: the Puiseux characteristic. The intermediate key result to lend this notion with topological weight is that a branch can be completely described through its associated knot, arising from the intersection of the branch with a small enough 3-sphere. The combination of these above-mentioned facts will then culminate in the project's most powerful result, which assures that two curves are equivalent if and only if their branches share the same Puiseux characteristics and intersection numbers.

Figures (8)

• Figure 1: A Newton Polygon
• Figure 2: Representation of $\phi$ with respect to the $y_j$ (red) and $\tilde{y}_j$ (blue) for $n=6$.
• Figure 3: the curve $C$ given by $x^2-y^3=0$, and the intersection $C\cap S_\epsilon$ shown in black.
• Figure 4: Representation of the knot $x=\epsilon e^{2i\theta}, y=\epsilon^{3/2}e^{3i\theta}$ in the torus.
• Figure 5: the carousel of the knot given by the branch $x=t^4, y=t^6+t^7$.

Theorems & Definitions (138)

• Definition 1.1.1
• Proposition 1.1.2
• Definition 1.1.3
• Example 1.1.4
• Remark 1.1.5
• Definition 1.1.6
• Remark 1.1.7
• Theorem 1.2.1: Newton's algorithm
• Remark 1.2.2
• proof : Proof of Theorem \ref{['thm:newton']}