Vol. 7, No. 6, 2013

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On the ample cone of a rational surface with an anticanonical cycle

Robert Friedman

Vol. 7 (2013), No. 6, 1481–1504
Abstract

Let Y be a smooth rational surface, and let D be a cycle of rational curves on Y that is an anticanonical divisor, i.e., an element of |KY |. Looijenga studied the geometry of such surfaces Y in case D has at most five components and identified a geometrically significant subset R of the divisor classes of square  2 orthogonal to the components of D. Motivated by recent work of Gross, Hacking, and Keel on the global Torelli theorem for pairs (Y,D), we attempt to generalize some of Looijenga’s results in case D has more than five components. In particular, given an integral isometry f of H2(Y ) that preserves the classes of the components of D, we investigate the relationship between the condition that f preserves the “generic” ample cone of Y and the condition that f preserves the set R.

Keywords
rational surface, anticanonical cycle, exceptional curve, ample cone
Mathematical Subject Classification 2010
Primary: 14J26
Milestones
Received: 2 August 2012
Revised: 27 November 2012
Accepted: 3 January 2013
Published: 19 September 2013
Authors
Robert Friedman
Department of Mathematics
Columbia University
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New York, NY
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United States