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

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.