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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

international journal of mathematics and mathematical sciences

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

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