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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
