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:

- $n = 2$: 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?