Classification of log del~Pezzo surfaces of index two
Vol. 14 (2007), No. 3, Page 293-498.
Noboru Nakayama
Classification of log del~Pezzo surfaces of index two
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Abstract:
In this article, a log del~Pezzo surface of index two means a projective normal non-Gorenstein surface \( S \) such that \( (S, 0) \) is a log-terminal pair, the anti-canonical divisor \( -K_{S} \) is ample and that \( 2K_{S} \) is Cartier. The log del~Pezzo surfaces of index two are shown to be constructed from data \( (X, E, \Delta) \) called fundamental triplets consisting of a non-singular rational surface \( X \), a simple normal crossing divisor \( E \) of \( X \), and an effective Cartier divisor \( \Delta \) of \( E \) satisfying a suitable condition. A geometric classification of the fundamental triplets gives a classification of the log del~Pezzo surfaces of index two. As a result, any log del~Pezzo surface of index two can be described explicitly as a subvariety of a weighted projective space or of the product of two weighted projective spaces. This classification does not use the theory of K3 lattices, which is essential for the classification by Alexeev--Nikulin \cite{AN}. The comparison between two classifications is also discussed.
Mathematics Subject Classification (2000): Primary 14J25, 14J26; Secondary 14J70, 14E30
Mathematical Reviews Number: MR2372472
Received: 2006-11-17
