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. 1
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).