Covering gonality of symmetric products of curves and Cayley–Bacharach condition on Grassmannians

Given an irreducible projective variety X, the covering gonality of X is the least gonality of an irreducible curve E ⊂ X passing through a general point of X. In this paper, we study the covering gonality of the k-fold symmetric product C(k) of a smooth complex projective curve C of genus g ≥ k + 1. It follows from a previous work of the first author that the covering gonality of the second symmetric product of C equals the gonality of C. Using a similar approach, we prove the same for the -fold and the -fold symmetric product of C. A crucial point in the proof is the study of the Cayley-Bacharach condition on Grassmannians. In particular, we describe the geometry of linear subspaces of Pn satisfying this condition, and we prove a result bounding the dimension of their linear span.