Description
The main functions concerning surfaces are:
This realizes the djunction process of Sommese and Van de Ven for smooth projective surfaces. Given a smooth projectiv surface X in Pn, Sommese and Van de Ven study X via the morphism phi=phi_{|H+K|}: X -> X1 subset Pn1 where H denotes the hyperplane class and K the canonical divisor on X.
Theorem[SVdV]. Let X subset Pn be a smooth projective curve. Then phi is a birational map onto a smooth projective surface X1, which blows down all (-1) lines E of X, i.e., (-1) curves of X which are embedded into Pn as lines, and is regular otherwise, unless (1) X is a P2 linearly or quadratically embedded, or ruled in lines, (2) X is a anticanonical embedded del Pezzo surface, (3) X is a conic bundle in which case phi_{|H+K|}: X -> B maps X to a curve B and the fibers are conics, or (4) X is an element of one of the four families of exceptions where |K+H| defines a finite to 1 map. See
specialFamiliesOfSommeseVandeVen
Since a (-1) conic on X becomes (-1) line on X1, repeating this process finally reaches a minimal surface unless X has negative Kodaira dimension.
In case of rational surfaces, repeating the process the process we end in one of the exceptional cases (1),..,(4). Whether we are in case (3) or (4) can be detect from the presentation matrix of O_X(H+K). If this presentation matrix is not linear, then a smooth hyperplane section C in |H| will be a hyperelliptic curves by Green's K_{p,1} theorem and phi is no longer birational, because it induces the canonical system on C which is 2:1. In our function adjunctionProcess we stop with this last surface.
References:
[DES] W. Decker, L. Ein, F.-O. Schreyer. Construction of surfaces in P4. J. Alg. Geom. 2 (1993), 185-237.
[SVdV] A. Sommese, A. Van de Ven. On the adjunction mapping, Math. Ann. 278 (1987), 593–603.
[G] M.Green. Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125-167, 168-171 (1984).