On irreducible germs of generic morphisms

Key Points

• The research aims to explore irreducible germs of generic morphisms to understand the singularities of their branch curves.
• Examined the set of irreducible germs $f_P$ associated with generic morphisms $f: S \to \mathbb P^2$.
• Analyzed branch curve germs $B_P \subset V_p$ with singularities characterized by coprime integers $k_1$ and $k_2$.
• Utilized the concept of equisingular deformation equivalence to draw conclusions about these singularities.
• Identified that the singularities of the branch curves relate to those defined by $x^{k_1}-y^{k_2}=0$ with coprime $k_1$ and $k_2$.
• Established a framework for understanding the deformation equivalence between different singularities in projective geometry.