Exceptional Loci and Hyperbolicity of Complements of Plane Curves with ${\bar{c}}_1^2-{\bar{c}}_2>0$

Let B be a plane curve which has simple normal crossings. We prove that, if B = B_1 \cup B_2 has two irreducible components with deg B_1, deg B_2 \geq 5 or deg B_1 = 4, deg B_2 \geq 7, and if B is general, then the associated algebraic exceptional set is empty. Based on this, we show that for a general plane curve B satisfying \bar{c}_1^2 - \bar{c}_2 > 0, the open surface $\mathbb{P}$^2 \setminus B is Kobayashi hyperbolic.