Variations on the Weak Bounded Negativity Conjecture

We present two applications of Hao’s proof of the Weak Bounded Negativity Conjecture. First, we address the so-called Weighted Bounded Negativity Conjecture and we prove that all but finitely many reduced and irreducible curves C on the blow-up of ℙ2 at n points satisfy the inequality $\begin{array}{} \displaystyle C^2 \ge \min \bigl\{-\frac{1}{12} n (C.L +27), -2 \bigr\}, \end{array}$ where L is the pull-back of a line. Next, we turn to the widely open conjecture that the canonical degree C.KX of an integral curve on a smooth projective surface X is bounded from above by an expression of the form A(g − 1) + B, where g is the geometric genus of C and A, B are constants depending only on X. We prove that this conjecture holds with A = − 1 under the assumptions h 0(X, −KX ) = 0 and h 0(X, 2KX + C) = 0.