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

Friedman, Robert

doi:10.2140/ant.2013.7.1481

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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

DOI: 10.2140/ant.2013.7.1481

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 $| - K_{Y} |$ . 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 $H^{2} (Y)$ 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 2990 Broadway New York, NY 10027 United States

Vol. 7, No. 6, 2013