Noncommutative blowups of elliptic algebras

We develop a ring-theoretic approach for blowing up many non commutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g ∈ T_1, T /gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex.