A characterization of double covers of curves in terms of the ample cone of second symmetric product

We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus g. This 2-dimensional nef cone gives a characterization of double covers of curves of genus image. This is a generalization of a result by Debarre [Olivier Debarre, Seshadri constants of abelian varieties, in: The Fano Conference, Univ. Torino, Turin, 2004, pp. 379–394, Proposition 8].