On irreducible germs of generic morphisms

TL;DR

This work studies irreducible germs of generic morphisms $f:S\to \mathbb{P}^2$ whose branch curve germs have equisingular $\,\mathcal{P}$-simplest singularities, tying local germ data to global monodromy and resolution graphs. It develops a combinatorial–Diophantine framework combining Euclid-type orbit analysis, continued fractions, and weighted chains to classify germ families $\mathcal{G}_{(2),\mathcal{O}}$ and $\mathcal{G}_{(2),\overline{\mathcal{D}}}$, and to describe the local-to-global monodromy constraints. The main results give explicit, unique (up to equivalence) germ realizations for each orbit type $\{k_1,k_2\}$ and each lift in $\mathcal{D}_{\mathcal{P},0}$, and prove that $\mathcal{G}_{(2),\mathcal{P}}$ splits as a disjoint union $\mathcal{G}_{(2),\mathcal{O}}\cup\mathcal{G}_{(2),\overline{\mathcal{D}}}$, with a Chisini-type theorem (constant $\frak d=12$) holding for extra-quasi-generic covers. The analysis combines resolution theory for $\,A_{k,p}$ cyclic quotient singularities, local fundamental groups, and monodromy to achieve a complete classification and to elucidate how singularity data governs the global cover structure and equivalence classes. This advances understanding of how branch-singularity structure controls the global geometry of plane covers and provides a rigorous bridge between singularity theory, topological monodromy, and algebraic classification.

Abstract

The article examines a set of irreducible germs $f_P:U_P\to V_p$ of %finite generic morphisms $f:S\to\mathbb P^2$ to the projective plane whose branch curve germs $B_P\subset V_p$ have singularities equisingular deformation equivalent to singularities given by equations $x^{k_1}-y^{k_2}=0$ with coprime $k_1,k_2\in\mathbb N$.

Theorems & Definitions (21)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Proposition 1
• Proposition 2
• Lemma 2