Abstract:  The Cox ring of a smooth projective variety X with finitely generated free Picard group is a universal coordinate ring graded by the Picard group of X and the support of this grading is the monoid of effective divisors on X. To provide a presentation of the Cox ring, or equivalently to give coordinates on the universal torsor, is a difficult problem as often many generators and relations are needed. We take a novel approach by replacing the universal torsor with an A^1-homotopic affine variety. Furthermore, we prove that the Cox ring of X is the ring of invariants for an additive group action on an affine variety. Then, using some ideas from non-reductive geometric invariant theory, we give criteria for effectivity of divisor classes on X. This is joint work with Brent Doran.