# Exercise about an algebraic surface

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 $g=g(C)$?

• Is the intersection of four smooth quadrics in $P^6$ automatically smooth? – Mariano Suárez-Álvarez May 26 '15 at 9:12
• yes i suppose the quadrics to be smooth – dario May 26 '15 at 9:17
• @dario : You did not answer Mariano's question – Patrick Da Silva May 29 '15 at 12:41
• The quadrics may be smooth, dario, but it doesn't follow that $S$ is smooth. I think you want to say that $S$ is a complete intersection of the $Q_i$. – Rhys May 29 '15 at 12:41
• Yes. that is the assumption. – dario May 29 '15 at 13:03

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.
• how can i compute the pluri genus $p_g$ and the irregurality $q$ of $S$ where $g$ is the genus of $C$? – dario May 27 '15 at 10:02