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?

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    $\begingroup$ What do you mean with "degree" for a morphism between varieties of different dimension? $\endgroup$
    – Wojowu
    May 27 at 11:12
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    $\begingroup$ @Wojowu The morphism $C\to\mathbb{P}^n$ determines a divisor on $C$, whose dimension is what I mean. $\endgroup$
    – Li Li
    May 27 at 11:24
  • $\begingroup$ I suppose you mean the degree of this divisor (I don't know what is the dimension of a divisor). $\endgroup$
    – abx
    May 27 at 13:11
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    $\begingroup$ 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. $\endgroup$
    – abx
    May 27 at 13:23
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    $\begingroup$ I am not sure that I understand the question. Every smooth curve embeds in $\mathbb{P}^3$, so the only non-trivial cases are $n=1, \, 2$. $\endgroup$ May 27 at 13:36

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