Blowup–blowdown classification conjecture for elliptic algebras

Let $T$ and $T'$ be elliptic algebras with equal graded quotient rings

$$Q_{\operatorname{gr}}(T)=Q_{\operatorname{gr}}(T').$$

A blowup or blowdown is the corresponding birational operation on elliptic algebras described in the paper. Blowup–blowdown classification conjecture. Up to isomorphism, $T$ and $T'$ are related by a finite series of blowdowns and blowups. The paper proves minimality and contraction results for several classes of elliptic algebras, but does not establish this full birational classification.