We have computed with divisors on curves for a decade. Analogous calculations are possible using curves on surfaces (rather than points on curves) with many new phenomena. Del Pezzo surfaces are the 2-dimensional analogues of conic curves. Their divisor class groups are lattices Z^d, where the bilinear form comes from the intersection of curves on a surface. The basic geometrical theory is available in magma, and I want to explain that and then show how it relates to some arithmetic properties of surfaces. This has plenty of relations to what Martin and Steve have been doing recently, to Mike’s sheaf code and to John Cremona’s code for conic curves over a function field; for comparison later, Andrew is working on similar theory but with an added finite group action.