How to write down a generic genus $g$ curve in $\mathbb{P}^n$ as an intersection of hypersurfaces? Let $g, n \geq 1$ be positive integers. How can we describe a generic genus $g$ curve inside $\mathbb{P}^n$ as an intersection of hypersurfaces?
For example when $g = 1$ and $n = 2,3,4$, we have the following descriptions:

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*$n = 2$: a generic genus 1 curve is given by a plane cubic;

*$n = 3$: a generic genus 1 curve is given by the intersection of two quadrics;

*$n = 4$: a generic genus 1 curve is given by the intersection of the five quadrics defined by $4 \times 4$ sub-Pfaffians of a $5 \times 5$ skew-symmetric of linear forms.

As one can see, the $(g,n) = (1,4)$ case becomes complicated, as a generic genus one curve can no longer be expressed as a complete intersection.
For $g = (d-1)(d-2)/2$, we always have a nice description of genus $g$ curves in $\mathbb{P}^2$: in particular, a smooth plane curve of degree $d$ will have genus $g$.
Are there any other pairs of $(g,n)$ which admit an explicit description of generic genus $g$ curves as the intersection (possibly not complete) of hypersurfaces?
 A: For $(g,n)=(g,g-1)$, i.e. the case of curves embedded by their complete canonical system for $g\geq 3$, one has a good understanding of the equations of a general $C$ for $g$ up to about $9$. The cases $g=3,4,5$ are classical and already described in the comments. The breakthrough was I believe Mukai's Curves and symmetric spaces I,II ($g=7,8,9$); there is a lot of followup work such as Ide-Mukai Canonical curves of genus 8, von Bothmer's Geometric Syzygies of Mukai Varieties and General Canonical Curves with Genus at most 8, etc. The main result (quoted from the last paper) is as follows.
Theorem (Mukai). Every general canonical curve of genus $7\leq g \leq 9$ is a general linear section of an embedded rational homogeneous variety $M_g$. General canonical curves of genus $6$ are cut out by a general quadric on a general linear section of a homogeneous variety $M_6$.
The equations of these homogeneous varieties can be written down reasonably explicitly using representation theory.
In a different direction, for $(g,n)=(1,m-1)$, one is talking about elliptic normal curves of degree $m$. Their equations are studied for example in Fisher's Pfaffian representations of elliptic normal curves, which generalises your examples for $g=1$.
For more general $(g,n)$ lots of things can happen. I recommend Eisenbud's A Mystery Variety in $P^3$ in the collection Computations in algebraic geometry with Macaulay 2.
