  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 g of degree 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. We also show that the ideal structure of T(d)
is quite rigid. Our results generalise those of the first author. In the
companion paper "Classifying Orders in the Sklyanin Algebra", we apply our
results to classify orders in (a Veronese subalgebra of) a generic cubic or
quadratic Sklyanin algebra.
