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

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Abstract Given an irreducible projective variety X, the covering gonality of X is the least gonality of an irreducible curve $E\subset X$ 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 $g\geq k+1$ . 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 $3$ -fold and the $4$ -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 $\mathbb {P}^n$ satisfying this condition, and we prove a result bounding the dimension of their linear span.