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 Nef (C(2))R of all numerically effective R-divisors classes in the Néron-Severi space N1(C(2))R. In a recent work, Julius Ross improved the bounds on Nef (C(2))R 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. © 2009 Springer-Verlag.