  In this paper we study the behaviour of critical points of the
Ginzburg-Landau perturbation of the Dirichlet energy into the sphere
$E_\varepsilon(u):=\int_\Sigma \frac{1}{2}|du|^2_h\ \,dvol_h
+\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_{\Sigma}e_{\varepsilon}(u)$.
Our first main result is a precise point-wise estimate for $e_\varepsilon(u_k)$
in the regions where compactness fails, which also implies the $L^{2,1}$
quantization in the bubbling process. Our second main result consists in
applying the method developed in a previous joint paper with T. Rivi\`ere to
study the upper-semi-continuity of the extended Morse index to sequences of
critical points of $E_{\epsilon}$: given a sequence of critical points
$u_{\varepsilon_k}:\Sigma\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that
converges in the bubble tree sense to a harmonic map $u_\infty\in
W^{1,2}(\Sigma,{S}^{n})$ and bubbles $v^i_{\infty}:\mathbb{R}^2\to {S}^{n}$, we
show that the extended Morse indices of the maps $v^i,u_\infty$ control the
extended Morse index of the sequence $u_{\varepsilon_k}$ for $k$ large enough.
