  In this paper we rigorously investigate the emergence of defects on Nematic
Shells with genus different from one. This phenomenon is related to a non
trivial interplay between the topology of the shell and the alignment of the
director field. To this end, we consider a discrete $XY$ system on the shell
$M$, described by a tangent vector field with unit norm sitting at the vertices
of a triangulation of the shell. Defects emerge when we let the mesh size of
the triangulation go to zero, namely in the discrete-to-continuum limit. In
this paper we investigate the discrete-to-continuum limit in terms of
$\Gamma$-convergence in two different asymptotic regimes. The first scaling
promotes the appearance of a finite number of defects whose charges are in
accordance with the topology of shell $M$, via the Poincar\'e-Hopf Theorem. The
second scaling produces the so called Renormalized Energy that governs the
equilibrium of the configurations with defects.
