Remarks on the nef cone on symmetric products of curves

Let C be a very general curve of genus g and let C(2) be its second symmetric product. This paper concerns the problem of describing the convex cone of all numerically effective R-divisors classes in the Néron–Severi space of C(2). In a recent work, Julius Ross improved the bounds on the nef cone of C(2) in the case of genus five. By using his techniques and by studying the gonality of the curves lying on C(2), we give new bounds on the nef cone of C(2) when C is a very general curve of genus 5 ≤ g ≤ 8.