Edited sentence: For a rational normal curve] it is the Eagon-Northcott complex.For elliptic curves in projective space the resolution may be constructed as a mapping cone of Eagon-Northcott complexes.In other words weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution.There are examples where the minimal free resolution is known explicitly