Minimal degree of the 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?

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?

• What do you mean with "degree" for a morphism between varieties of different dimension? May 27 at 11:12
• @Wojowu The morphism $C\to\mathbb{P}^n$ determines a divisor on $C$, whose dimension is what I mean. May 27 at 11:24
• I suppose you mean the degree of this divisor (I don't know what is the dimension of a divisor).
– abx
May 27 at 13:11
• If I understand correctly the question, the answer is obviously yes. You have a base-point free linear system $\lvert D\rvert$ on $C$ of (projective) dimension $n$ and degree $d$; for general points $p_1,\ldots ,p_{n-1}$ in $C$, the linear system $\lvert D-p_1-\ldots -p_{n-1}\rvert$ has dimension 1 and degree $d-(n-1)$, hence $\operatorname{gon}(C)\leq d- (n-1)$ and $d\geq \operatorname{gon}(C)+n-1$, which is then the minimum possible.
– abx
May 27 at 13:23
• I am not sure that I understand the question. Every smooth curve embeds in $\mathbb{P}^3$, so the only non-trivial cases are $n=1, \, 2$. May 27 at 13:36