Exercise about an algebraic surface

Question

Let $\mathbb{P}^6$ the six-dimensional complex projective space. Suppose that $Q_{i}$ is a smooth quadric in $\mathbb{P}^6$ for $i=1,...,4$. Define $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4 $$ as complete intersection of four quadric. So, by definition, $S$ is a smooth algebraic surface in $\mathbb{P}^6$. Let $H$ an hypersurface in $\mathbb{P}^6$. Put $C=S \cap H$.
Question:
How can i compute all numerical invariant of the surface $S$ i.e. $K_S^2$, $\chi(O_S)$, $p_g$, $q$ and so on ? is there a formula to compute $g=g(C)$?

Answer 1

For (2), use adjunction formula and get \begin{eqnarray} K_S=i^*_S(K_{\mathbb{P}^6}\otimes\mathcal{O}_{\mathbb{P}^6}(2)^{\otimes 4})=i_S^*(\mathcal{O}_{\mathbb{P}^6}(-7)\otimes\mathcal{O}_{\mathbb{P}^6}(2)^{\otimes 4})=\mathcal{O}_S(1) \end{eqnarray} Relation between $K_S$ and $K_C$ can also be obtained using adjunction formula.