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 ℙ 2 at n points satisfy the inequality array C² ≥ min \-1/12 n (C.L +27), -2 \, array where L is the pull-back of a line. Next, we turn to the widely open conjecture that the canonical degree C . K X 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 , − K X ) = 0 and h 0 ( X , 2 K X + C ) = 0.