Exceptional curves on smooth rational surfaces with-𝐾 not nef and of self-intersection zero
M Lahyane - Proceedings of the American Mathematical Society, 2005 - ams.org
Proceedings of the American Mathematical Society, 2005•ams.org
A $(-n) $-curve is a smooth rational curve of self-intersection $-n $, where $ n $ is a positive
integer. In 1998 Hirschowitz asked whether a smooth rational surface $ X $ defined over the
field of complex numbers, having an anti-canonical divisor not nef and of self-intersection
zero, has $(-2) $-curves. In this paper we prove that for such a surface $ X $, the set of $(-1)
$-curves on $ X $ is finite but non-empty, and that $ X $ may have no $(-2) $-curves. Related
facts are also considered. References
integer. In 1998 Hirschowitz asked whether a smooth rational surface $ X $ defined over the
field of complex numbers, having an anti-canonical divisor not nef and of self-intersection
zero, has $(-2) $-curves. In this paper we prove that for such a surface $ X $, the set of $(-1)
$-curves on $ X $ is finite but non-empty, and that $ X $ may have no $(-2) $-curves. Related
facts are also considered. References
Abstract
A -curve is a smooth rational curve of self-intersection , where is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has -curves. In this paper we prove that for such a surface , the set of -curves on is finite but non-empty, and that may have no -curves. Related facts are also considered. References
ams.org