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On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces


We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

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The authors are indebted to Hiromu Tanaka for the very useful discussion and informing them of the current state of the art of the base point free theorem on surfaces. They also thank Kazuhiro Konno for informing them of the works by Noboru Nakayama and Yoichi Miyaoka. S. O.  was partially supported by Grants-in-Aid for Scientific Research (16H05994, 16K13746, 16H02141, 16K13743, 16K13755, 16H06337) and the Inamori Foundation.

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Correspondence to Rikito Ohta.

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Ohta, R., Okawa, S. On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces. manuscripta math. 166, 1–17 (2021).

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Mathematics Subject Classification

  • 14J26
  • 14E30
  • 14G17