Let $K$ be an algebraically closed field. I'm interested in isomorphism classes of triples $(V,f,g)$ where $V$ is a finite dimensional $K$-vector space and $f,g$ are commuting endomorphisms of $V$. What I expect is a "normal form" theorem, similar to Jordan's one - which solves the case of a single endomorphism.

If we can find some smooth irreducible curve $\, \mathcal{C} \, : \, P(x,y) = 0$ defined over $K$ such that $P(f,g) = 0$, then the $K[X,Y]$-module structure on $V$ factors through the coordinate ring of $\mathcal{C}$, which is a Dedekind domain. One can then apply the "structure theorem for finitely generated modules over Dedekind domains", which yields : $(V,f,g)$ has a (essentially unique) direct sum decomposition into triples of the form $(K[X,Y]/(X,Y)^n,\alpha + X, \beta + Y)$, where $n \geq 1$ and $(\alpha,\beta)$ is some point of $\mathcal{C}$.

Is there a similar result in absence of such a nice curve ? (This question was raised in the comments of this mathstackexchange question.)