The interpolation conjecture for Brill–Noether curves

Let $d$, $g$, $r$, and $n$ be nonnegative integers with

$$\rho(d,g,r)=(r+1)d-rg-r(r+1)\geq 0.$$

A BN-curve is a stable map from a general curve of genus $g$ to $\mathbb{P}^r$ of degree $d$ corresponding to the unique component dominating $\overline{M}_g$. The interpolation conjecture. There is a BN-curve of degree $d$ and genus $g$ through $n$ general points in $\mathbb{P}^r$ if and only if

$$(r-1)n\leq (r+1)d-(r-3)(g-1),$$

apart from finitely many exceptions. This conjecture proposes that the evident dimension inequality usually suffices for interpolation, despite known cases where it does not.