Algebra & Number Theory

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

Robert Friedman

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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.

Article information

Source
Algebra Number Theory, Volume 7, Number 6 (2013), 1481-1504.

Dates
Received: 2 August 2012
Revised: 27 November 2012
Accepted: 3 January 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730035

Digital Object Identifier
doi:10.2140/ant.2013.7.1481

Mathematical Reviews number (MathSciNet)
MR3107570

Zentralblatt MATH identifier
06226676

Subjects
Primary: 14J26: Rational and ruled surfaces

Keywords
rational surface anticanonical cycle exceptional curve ample cone

Citation

Friedman, Robert. On the ample cone of a rational surface with an anticanonical cycle. Algebra Number Theory 7 (2013), no. 6, 1481--1504. doi:10.2140/ant.2013.7.1481. https://projecteuclid.org/euclid.ant/1513730035


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