Let \(X\) be a surface in \(\PP^n\), \(n \geqslant 3\), 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.
Solution
Write \[ H=\OO_X(1) \] for the hyperplane class and put \[ a=\sum_{i=1}^{n-2}d_i-n-1. \]
1. The canonical class of \(X\) is\[ \boxed{K_X\sim aH}. \]
For a nonsingular complete intersection of hypersurfaces of degrees \(d_1,\ldots,d_{n-2}\) in \(\PP^n\), the adjunction formula P-AGH284COMPINT gives \[ \omega_X \cong \OO_X\left(\sum_{i=1}^{n-2}d_i-n-1\right). \] With the notation fixed above, this is \[ \omega_X\cong\OO_X(a), \] or equivalently \[ K_X\sim aH. \]
2. If \(a>0\), then \(X\) is of general type.
The hyperplane class \(H\) is ample on \(X\). Hence for \(a>0\) the canonical divisor \[ K_X\sim aH \] is ample. Every ample divisor is big, so \(K_X\) is big. Therefore \(X\) is of general type.
3. If \(X\) is not covered by step 2, then, up to permuting the\(d_i\), its degree data are exactly\[ \boxed{ \left\{ (3;2),\, (3;3),\, (3;4),\, (4;2,2),\, (4;2,3),\, (5;2,2,2) \right\}. } \]
The exceptional condition is \[ a\le0, \] equivalently \[ \sum_{i=1}^{n-2}d_i\le n+1. \] Since every \(d_i\ge2\) and there are \(n-2\) of them, \[ \sum_{i=1}^{n-2}d_i\ge2(n-2)=2n-4. \] Thus an exceptional tuple must satisfy \[ 2n-4\le n+1, \] so \[ n\le5. \]
For \(n=3\) there is one degree \(d_1\ge2\), and \[ d_1\le4, \] giving \[ d_1=2,3,4. \]
For \(n=4\) there are two degrees at least \(2\) with \[ d_1+d_2\le5. \] Up to order, the only possibilities are \[ (d_1,d_2)=(2,2),(2,3). \]
For \(n=5\) there are three degrees at least \(2\) with \[ d_1+d_2+d_3\le6, \] so necessarily \[ (d_1,d_2,d_3)=(2,2,2). \] There are no possibilities for \(n\ge6\).
4. The three exceptional tuples with \(a<0\) are del Pezzo surfaces:\[ \begin{array}{c|c|c|c} (n;d_i) & K_X & K_X^2 & \text{surface}\\ \hline (3;2) & -2H & 8 & \text{quadric surface}\\ (3;3) & -H & 3 & \text{cubic surface}\\ (4;2,2) & -H & 4 & \text{intersection of two quadrics}. \end{array} \]
For each of these tuples step 1 gives \(-K_X\) as a positive multiple of the ample hyperplane class. Hence \(-K_X\) is ample, so \(X\) is a del Pezzo surface.
For a complete-intersection surface, \[ H^2=\deg X=\prod_i d_i. \] Consequently \[ K_X^2=a^2H^2. \] For the quadric in \(\PP^3\), one has \(a=-2\) and \(H^2=2\), so \[ K_X^2=8. \] For the cubic in \(\PP^3\), one has \(a=-1\) and \(H^2=3\), so \[ K_X^2=3. \] For the \((2,2)\) complete intersection in \(\PP^4\), one has \(a=-1\) and \(H^2=4\), so \[ K_X^2=4. \]
Thus these are the del Pezzo surfaces of degrees \(8\), \(3\), and \(4\), respectively. Over the algebraically closed ground field, the smooth quadric is \(\PP^1\times\PP^1\), while the cubic surface and the degree-\(4\) del Pezzo surface are the familiar rational del Pezzo surfaces. Hence these cases lie in the \(\kappa=-\infty\) part of the surface classification.
5. The three exceptional tuples with \(a=0\) are K3 surfaces:\[ \begin{array}{c|c} (n;d_i) & H^2\\ \hline (3;4) & 4\\ (4;2,3) & 6\\ (5;2,2,2) & 8. \end{array} \]
For these three tuples, \[ \sum_i d_i=n+1, \] so step 1 gives \[ \omega_X\cong\OO_X. \]
The complete-intersection cohomology calculation P-AGH355COMPINT applies to a surface and gives \[ H^1(X,\OO_X)=0. \] It also gives \[ H^0(X,\OO_X)=k. \] Hence \[ q(X)=0 \] and, since \(\omega_X\cong\OO_X\), \[ p_g(X) = h^0(X,\omega_X) = h^0(X,\OO_X) =1. \] Thus these are K3 surfaces in the surface-classification picture, and they lie in the \(\kappa=0\) class.
Their indicated polarization degrees are simply \[ H^2=\deg X=\prod_i d_i, \] namely \(4\), \(6\), and \(8\).
6. All other degree data give surfaces of general type, while the sixtuples in step 3 are precisely the del Pezzo and K3 exceptions listedin steps 4 and
5.
Step 3 exhausts every tuple for which \(a\le0\). Therefore every tuple not in that list has \(a>0\), and step 2 makes \(X\) of general type. Among the six exceptions, step 4 classifies the three cases with \(a<0\), and step 5 classifies the three cases with \(a=0\).
7. Q.E.D.
Step 1 computes the canonical class. Steps <1>2–<1>3 reduce the problem to exactly six exceptional degree patterns. Steps <1>4–<1>5 place those exceptions in the del Pezzo and K3 parts of the surface classification, and step 6 proves that every remaining case is of general type.