Title: Complements of quartic plane curves and the Geometric Lang-Vojta Conjecture
Abstract: Given a very general quartic curve X in the plane, we ask the following natural question: what is the minimum number of times that a degree d curve of genus g can meet it? In this talk, we give a bound on this number in terms of d and g, and show that this bound is sharp for conics. This shows that the complement of a very general quartic plane curve is algebraically hyperbolic outside the locus of flex and bitangent lines, resolving the last open case of the algebraic hyperbolicity of the complement of very general plane curves. In the talk, we describe how algebraic hyperbolicity is a natural analogue of the number theory and complex analysis notions of "hyperbolic," and show how our work proves the Geometric Lang-Vojta Conjecture for the complements of plane curves. This is joint with Xi Chen and Wern Yeong.