#### 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 $|-{K}_{Y}|$. 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 $\left(Y,D\right)$, 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 ${H}^{2}\left(Y\right)$ 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
Primary: 14J26