## V.6: Classification of Surfaces 


### V.6.1.  #to-work

Let $X$ be a surface in $\mathbf{P}^n, n \geqslant 3$, defined as the complete intersection of hypersurfaces of degrees $d_1, \ldots, d_{n-2}$, with each $d_i \geqslant 2$. Show that for all but finitely many choices of $\left(n, d_1, \ldots, d_{n-2}\right)$, the surface $X$ is of general type. List the exceptional cases, and where they fit into the classification picture.

### V.6.2.  #to-work

Prove the following theorem of Chern and Griffiths. Let $X$ be a nonsingular surface of degree $d$ in $\mathbf{P}_{\mathbf{C}}^{n+1}$, which is not contained in any hyperplane. If $d<2 n$, then $p_g(X)=0$. If $d=2 n$, then either $p_g(X)=0$, or $p_g(X)=1$ and $X$ is a K3 surface.[^hint.v.6.2]

[^hint.v.6.2]: 
Hint: Cut $X$ with a hyperplane and use Clifford's theorem (IV, 5.4). For the last statement, use the Riemann-Roch theorem on $X$ and the Kodaira vanishing theorem (III, 7.15).