Nonnormal del Pezzo surfaces

This paper studies reduced, connected, Gorenstein surfaces with ample -K, assumed to be reducible or nonnormal. The normalisation is a union of one or more standard surfaces (scrolls and Veronese surfaces), marked with a conic as double locus. The question is how to glue these together to get a Gorenstein scheme. In characteristic 0, the results amount to a classification of projective surfaces in the style of the 1880s. However, the methods involve a study of the dualising sheaf of a nonnormal variety in terms of Rosenlicht differentials, and there is a subtle pathology in characteristic p due to Mori and S. Goto.Comment: amsTeX 2.1 (amsppt format), submitted to Math Proceedings, RIM

Diptych varieties. I

We present a new class of affine Gorenstein 6-folds obtained by smoothing the 1-dimensional singular locus of a reducible affine toric surface; their existence is established using explicit methods in toric geometry and serial use of Kustin-Miller Gorenstein unprojection. These varieties have applications as key varieties in constructing other varieties, including local models of Mori flips of Type A.Comment: 50 pages. The webpage at www-staff.lboro.ac.uk/~magdb/aflip.html contains links to auxiliary materia