Factorization in noncommutative curves (Q1030761)

Let \(k\) be an algebraically closed field of characteristic zero. The author considers a commutative curve \(C=Z(f_0)\) over the field \(k\) and its noncommutative models \(\tilde{C}=Z(f)\). Given a polynomial \(f_0\in k[x,y]\) it has in general many different noncommutative models \(f\in k<x,y>\), ie a polynomial \(f\) with \(\pi(f)=f_0\). Here \(\pi:k<x,y>\rightarrow k[x,y]\) is the canonical map of rings. The author considers the following problem: Assume there is a factorization of \(f\) in \(k<x,y>\). What condition does this impose on the polynomial \(f_0\)? The author solves this problem in various situations.