Gian Pietro Pirola, Sections of the Jacobian bundles of plane curves and applications

We study normal functions (sections of the Jacobian bundle) defined on the moduli space of pointed plane curves. Using the infinitesimal Griffiths invariant (refined by M. Green and C. Voisin) we show that a normal function with nontrivial but sufficiently “small” support cannot be “locally constant”. As an application, we give a variational proof of the following result of Zu:

Theorem: If C is a very general plane curve of degree d and C’ is any plane curve of degree d’, then the cardinality i(C, C’) of the intersection between C and C’ is > d-3.

We also show that if d > 3 and i(C, C’) = d-2 then d’ = 1 and C’ is a bitangent or a flex line. For d = 4 this is a result of Chen, Rield and Yeong. This is a joint work with Lorenzo Fassina.