Ranks of sections of the Picard bundle of a family of curves

In this talk we study sections of the relative Picard bundle associated with a family of smooth curves of genus $g\geq 2$ via the rank of the associated normal function. Using Griffiths’ formula for the infinitesimal invariant together with higher-order Schiffer variations, we derive a numerical inequality relating the rank, the minimal support of a representing divisor, and the modular dimension of the family. When the modular map is dominant, we obtain a sharp classification of the equality case: equality occurs only for multiples of odd theta characteristics or for the canonical section. This is a joint work with Gian Pietro Pirola.