Let $k[x,y]$ be a polynomial ring in two indeterminants and $f \in k[x,y] \backslash k$ a non constant poynomial.
I want to know if there exist any nice criteria to answer the question when the quotient $R:=k[x,y]/(f)$ has the property that every localization at every maximal prime of $R$ is UFD.
The notation "nice" quantifies the generality of the application (preferably a wide class of different polynomials) and non triviality of the seeked criterion. eg a "boring" sufficient criterion is to require that $f$ contains a linear term $ax$ or $by$. Thus we can a priori assume that $f$ not contains linear terms.
The question is closely related to this one and considered as a far generalization of it: Localizations of $k[y,z]/(1-y^2+z^2)$ UFDs
Suppose that the condition you are looking for holds. Then the curve defined by $f$ is normal, since UFDs are integrally closed. But normalization is a resolution of singluarities in dimension 1, so if $k$ is, for example, algebraically closed, you end up with a smooth curve (and obviously the converse holds).