Optimal bound for the number of <mml:math alttext="$(-1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-curves on extremal rational surfaces

Lahyane, Mustapha

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International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 2, Pages 123-126
doi:10.1155/S0161171202013194

Optimal bound for the number of (−1)-curves on extremal rational surfaces

Mustapha Lahyane

Abdus Salam International Centre for Theoretical Physics, Trieste 34100, Italy

Received 14 May 2001; Revised 12 September 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give an optimal bound for the number of (−1)-curves on an extremal rational surface X under the assumption that −KX is numerically effective and having self-intersection zero. We also prove that a nonelliptic extremal rational surface has at most nine (−1)-curves.