Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$.
Put $C=H \cap S$ where $H$ is an hyperplane of $\mathbb{P}^6$. So $C$ is a smooth curve on $S$ with genus $g=g(C)$. Set $\eta_{C/S}$ the normal bundle. So you can show that $\eta_{C/S}$ satisfy $\omega_C=(\eta_{C/S})^{\otimes2}$ where $\omega_C$ is the canonical bundle on the curve $C$. So the normal bundle is a theta characteristic on $S$.
My questions:
1) what kind of theta-characteristic? odd or even?
2) suppose that $\mathscr{M}_S$ is the moduli space of the surface $S$ and $\mathscr{L}_C$ is the moduli space of the set $S_g=\{C, \theta_C \}$ where $\theta_C$ is a theta-characteristic on $C$. What are the dimensions of $\mathscr{M}_S$ and $\mathscr{L}_C$? is there a relations between these moduli spaces?
any suggestions are welcome.