On a question of Sean Keel
2011
Abstract Let X be a smooth projective surface defined over F p ¯ , and let L be a line bundle over X such that L ⋅ Y > 0 for every complete curve Y contained in X . A question of Keel asks whether L is ample. If X is a P 1 -bundle over a curve, we prove that this question has an affirmative answer.
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