Gorenstein models of del Pezzo surfaces of degree 1 over Dedekind schemes

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Abstract

Let R be a Dedekind scheme, \(\nu\) its generic point, X and V del Pezzo surfaces of degree 1 over R that are Gorenstein Mori fiber spaces (as 3-folds germs over the ground field). We study birational maps \(\phi:X\dasharrow V\) over R which are isomorphisms over the generic point of R. We put down normal forms of such transformations (in suitable coordinates) and give some properties of X and V. In particular, we prove the uniqueness of a smooth model.

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On the Classification of Cubic Surfaces

J. Bruce, C. Wall (1979)

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Nonnormal del Pezzo surfaces

Miles Reid (1994)

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.