ON THE COMPLEMENT OF A GENERIC CURVE IN THE PROJECTIVE PLANE

Abstract. Let C be a very generic smooth curve of degree d in the complex projective plane. In this paper, we show that for any curve D in the projective plane that does not contain C, the set C \ D contains at least d 2 distinct points. Moreover, this bound is sharp when d 3. Our motivation for the study of this problem comes from the problem of hyperbolicity of the complement of a plane curve. As an application of our result, we give another proof of the following fact: if C is a very generic smooth curve of degree d 5 in the complex projective plane, then the intersection of the complement of C in the projective plane with any plane curve D which does not contain C is hyperbolic. Let C be a generic smooth curve of degree d in the complex projective plane P2. We are interested in the following question: if D P2 is a curve with C 6 D, what is the least number of points that D \ C must contain? In this paper, we give a complete answer to this problem. THEOREM 1. If C is a very generic smooth curve of degree d in the complex projective plane P2, then for any curve D P2 with C 6 D, the set C\D contains at least d 2 distinct points. Moreover, this bound is sharp when d 3. Her