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?

Well, this is trivially false for $n$ at least $2g-1$, in which case the dimension is uniquely determined by the degree. Are there any non-trivial counter-examples, in the sense that the term $h^0(K-D)$ in Riemann-Roch does not vanish?

degreeof this divisor (I don't know what is the dimension of a divisor). $\endgroup$3more comments