Generators of Subcones of the Nef Cone of a Cubic Surface

● L = φ∗O(1) ● Li = L −Ei, the strict transform of a line through Pi = φ(Ei) ● Lij = 2L − (∑En) + Ei + Ej, the strict transform of a conic through the four points Pn with n ≠ i, j ● Bi = 3L− (∑En)−Ei, the strict transform of a cubic curve through all six points Pn, with a node at Pi. Let h be the class of a hyperplane in the embedding X ⊂ P3. For any line ` on X, the hyperplanes containing ` give (after removing `) a base-point-free pencil on X. The classes {Li, Lij,Bi} defined above are the 27 pencils coming from the lines. In addition, for any point x on X let Cx be the intersection of X with its tangent plane at x (so Cx has class h). If x does not lie on a line, then Cx is a plane cubic curve with one double point, at x. Let Γ be the nef cone of X, and S be the set of 27 divisor classes {Li, Lij,Bi} as i and j range over all possible values. For each element C in S, we define the subcone Γ(C) by: Γ(C) = {D ∈ Γ ∣D.C = min C′∈S {D.C ′}andD.C ⩽ (D.h)/2} . Further define the subcone Γ(h) to be: Γ(h) = {D ∈ Γ ∣ (D.h)/2 ⩽ min C′∈S {D.C ′}} .