Self-intersection of the canonical bundle of a projective variety

Fix integers n, r and d such that 2 ≤ n ≤ r − 1 and let V(n, r, d) be the set of all smooth, irreducible, projective, and nondegenerate varieties V of degree d and dimension n in the projective space P r , such that, only for n ≥ 3, the canonical bundle K V is numerically effective (i.e. K V ·C ≥ 0 for all curves C ⊂ V). Put K(n, r, d) = sup {K V n : V ε ν (n, r, d)}. In this paper we compute K(n, r, d) in terms of n, r and d and classify the varieties attaining the bound, at least when d ≫ r and for some range of n and r. In the case of surfaces, i.e. when n = 2, in the case r ≥ 2n + 1 and for d ≫ r we give a complete classification. It turns out that only in certain cases the varieties reaching the bound K (n, r, d) are Castelnuovo's varieties, that is varieties of maximal geometric genus.