In the first part of this article, we give bounds on self-intersections C² of integral curves C on blow-ups BlₙX of surfaces X with the anti-cannonical divisor -KX effective. In the last part, we prove the weak bounded negativity for self-intersections C² of integral curves C in a family of surfaces f:Y B where B is a smooth curve.
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Misra et al. (2024) studied this question.
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