The Orbibundle Miyaoka–Yau–Sakai Inequality and an Effective Bogomolov–McQuillan Theorem

Let X be a minimal projective surface of general type defined over the complex numbers and let C ⊂ X be an irreducible curve of geometric genus g . Given a rational number α \in [0, 1] , we construct an orbibundle \tilde {\mathcal E}_α associated with the pair (X, C) and establish the Miyaoka–Yau–Sakai inequality for \tilde {\mathcal E}_α . By varying the parameter α in the inequality, we derive several geometric consequences involving the “canonical degree” CK_X of C . Specifically we prove the following two results. (1) If K^2_X is greater than the topological Euler number c_2(X) , then CK_X is uniformly bounded from above by a function of the invariants g , K^2_X and c_2(X) (an effective version of a theorem of Bogomolov–McQuillan). (2) If C is nonsingular, then CK_X ≤ 3g − 3 + o(g) when g is large compared to K^2_X , c_2(X) (an affrmative answer to a conjecture of McQuillan).