  One of the difficulties in doing noncommutative projective geometry via
explicitly presented graded algebras is that it is usually quite difficult to
show flatness, as the Hilbert series is uncomputable in general. If the algebra
has a regular central element, one can reduce to understanding the (hopefully
more tractable) quotient. If the quotient is particularly nice, one can proceed
in reverse and find all algebras of which it is the quotient by a regular
central element (the filtered deformations of the quotient). We consider in
detail the case that the quotient is an elliptic algebra (the homogeneous
endomorphism ring of a vector bundle on an elliptic curve, possibly twisted by
translation). We explicitly compute the family of filtered deformations in many
cases and give a (conjecturally exhaustive) construction of such deformations
from noncommutative del Pezzo surfaces. In the process, we also give a number
of results on the classification of exceptional collections on del Pezzo
surfaces, which are new even in the commutative case.
