Inner rates of finite morphisms

Abstract

Let $(X, 0)$ be a complex analytic surface germ embedded in $(\mathbb{C}^n,0)$ with an isolated singularity and $Φ=(g,f):(X,0) \longrightarrow (\mathbb{C}^2,0)$ be a finite morphism. We define a family of analytic invariants of the morphism $Φ$, called inner rates of $Φ$. By means of the inner rates we study the polar curve associated to the morphism $Φ$ when fixing the topological data of the curve $(gf)^{-1}(0)$ and the surface germ $(X,0)$, allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function $f:(X,0) \longrightarrow (\mathbb{C},0)$. The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ $(X,0)$ and the curve $(gf)^{-1}(0)$.

Figures (8)

• Figure 1: The numbers between parenthesis are the orders of vanishing $(m_v(g),m_v(f))$ of the functions $g \circ \pi$ and $f \circ \pi$ along the irreducible components of $E$, these numbers can be determined from the dual graph using Proposition \ref{['laufer']}.
• Figure 2: The graph $\Gamma_{\pi}$, decorated with the orders of vanishing of the function $f \circ \pi$ and red arrows corresponding to the components of the polar curve weighted with the intersection numbers.
• Figure 3: The dual graph weighted with the inner rates $q_i$.
• Figure 4: The graph $\Gamma_{\pi}$ decorated with the orders of vanishing of the function $f \circ \pi$ and the length of each of its edges.
• Figure 5: The graph $\Gamma_{\pi}$ is weighted with the inner rates (without parenthesis) and the Hironaka quotients (between parenthesis).

Theorems & Definitions (89)

• Theorem A: The inner rates formula
• Theorem B
• Theorem : Michel2008
• Proposition C
• Theorem D: Theorem \ref{['Milnor']}
• Theorem E: Theorem \ref{['concentration']}
• Definition 1.1
• Definition 1.2
• Proposition 1.3: laufer1972 or nem for a topological proof
• Definition 1.4