# Indecomposable branched coverings over the projective plane by surfaces $M$ with $\chi(M) \leq 0$

## TL;DR

This paper investigates when branched coverings over the projective plane with surfaces of non-positive Euler characteristic are indecomposable, providing realizability results for certain data with even defect.

## Contribution

It extends known results by characterizing the realizability of branched coverings with odd degree and large defect, focusing on indecomposability over the projective plane.

## Key findings

- Realizability of data with even defect greater than degree for odd degree coverings.
- Extension of known results to cases where the degree is odd and the defect exceeds the degree.
- Conditions under which branched coverings are indecomposable over the projective plane.

## Abstract

In this work we study the decomposability property of branched coverings of degree $d$ odd, over the projective plane, where the covering surface has Euler characteristic $\leq 0$. The latter condition is equivalent to say that the defect of the covering is greater than $d$. We show that, given a datum $\mathscr{D}=\{D_{1},\dots,D_{s}\}$ with an even defect greater than $d$, it is realizable by an indecomposable branched covering over the projective plane. The case when $d$ is even is known.