# On the almost generic covers of the projective plane

## TL;DR

This paper investigates the singularities of branch curves and computes key invariants of almost generic covers of the projective plane, revealing their geometric properties and ramification behavior.

## Contribution

It characterizes the singular points of branch curves and derives invariants of the covering surface based on the branch curve's properties.

## Key findings

- Singular points of branch curves are classified.
- Main invariants of the surface are expressed via branch curve invariants.
- Properties of ramification locus are elucidated.

## Abstract

A finite morphism $f:X\to \mathbb P^2$ of a a smooth irreducible projective surface $X$ is called an almost generic cover if for each point $p\in \mathbb P^2$ the fibre $f^{-1}(p)$ is supported at least on $deg(f)-2$ distinct points and $f$ is ramified with multiplicity two at a generic point of its ramification locus $R$. In the article, the singular points of the branch curve $B\subset\mathbb P^2$ of an almost generic cover are investigated and main invariants of the covering surface $X$ are calculated in terms of invariants of the curve $B$.