Noncommutative Blowups of Elliptic Algebras

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈ T 1 , T / g T 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. We also show that the ideal structure of T ( d ) is quite rigid. Our results generalise those of the first author in Rogalski (Advances Math. 226 , 1433–1473, 2011 ). In the companion paper Rogalski et al. ( 2013 ), we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.