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Given an irreducible projective variety $X$, the covering gonality of $X$ isthe least gonality of an irreducible curve $E\subset X$ passing through ageneral 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 thatthe covering gonality of the second symmetric product of $C$ equals thegonality of $C$. Using a similar approach, we prove the same for the $3$-foldand the $4$-fold symmetric product of $C$. A crucial point in the proof is thestudy of Cayley-Bacharach condition on Grassmannians. In particular, wedescribe the geometry of linear subspaces of $\mathbb{P}^n$ satisfying thiscondition and we prove a result bounding the dimension of their linear span.

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