Minimal degree of a morphism from a curve to $\mathbb{P}^n$ We do all the things in an algebraically closed field $k$ of characteristic $0$. Let $C$ be a projective curve over $k$. We have been familiar with the notion "gonality", which is the minimal degree of a morphism from $C$ to $\mathbb{P}^1$. I'm curious about what we know about the minimal degree of a morphism $C\to\mathbb{P}^n$, for which in particular, I'm curious about whether the interested minimum for $C\to\mathbb{P}^n$ is exactly $\mathrm{gon}(C)+n-1$. It is obvious for rational curves and elliptic curves, but are there any results for general cases?
 A: Of course you can map the curve onto $\mathbb{P}^1$ and then embed $\mathbb{P}^1$ as a line into $\mathbb{P}^r$, but this would be not very significant, so it is better to restrict to nondegenerate maps $C\to \mathbb{P}^r$, meaning that the image is not contained in any hyperplane.
The space of nondegenerate maps $C\to \mathbb{P}^r$ of degree $d$ is essentially parametrized by the Brill-Noether variety $W^r_d(C)$, so you want to fix $r$ and find the smallest $d$ for which $W^r_d(C)\neq \emptyset$. If $C$ is a general curve of genus $g$, then we know that the Brill-Noether variety has dimension given by the Brill-Noether number.
$$ \dim W^r_d(C) = g - (r+1)(g+r-1) $$
In particular it is nonempty if and only if the number is positive. For example, if you take a general curve of genus $6$, we can see that it has gonality 4 but the minimum degree of a nondegenerate map $C\to \mathbb{P}^2$ is 6, which is one more than $5+1$. Observe that if $D$ is the divisor that gives the map to $\mathbb{P}^2$, then $h^0(D)=3$ and $h^0(K_C-D) = 2$ by Riemann-Roch.
