Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r   points in very general position on a smooth elliptic quartic curve B⊂P3B⊂P3 and let L∈Pic(X)L∈Pic(X) be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H0(X,L)H0(X,L).