It is well known that if a smooth curve $C \subset \mathbb{P}^3$ has degree $ d \leq 6$. Then 
$ g(C) \leq 4$ (Hartshorne pg 354). I know that the case $g=4$ correspond to the complete intersection of a quadratic and a cubic surface. Therefore, the ideal of the curve is
$$
I\_C = (f\_2(x\_0, x\_1,x\_2, x\_3),f\_3(x\_0, x\_1,x\_2, x\_3))
$$
I am wondering: Is it possible to find explicitly the ideal for the other genus? For example, what are the equation defining a rational curve in $\mathbb{P}^3$ of degree $6$ ? Is there a general strategy for finding those equations? or at least some bounds? What does it happen if we drop the smooth condition for admitting some mild singularities i.e $A\_1$


Thanks for any reference!



explicity equations for curves in the projective space
As explained by J.C. Ottem, the curves of degree $6$ are contained in a cubic. Let us assume that the surface is smooth, so you can see the cubic surface as the blow-up of six points in $\mathbb{P}^2$ with no $3$ collinear and not all on the same conic.


The surface $X$ has Picard group generated by $L$, the preimage of a line in $\mathbb{P}^2$, and by $E\_1,\dots,E\_6$, the exceptional divisors.


The trace of an hyperplane is the anti-canonical divisor, $-K\_X=3L-E\_1-\dots-E\_6$. 


$1)$ To obtain degree $6$ rational curves in $X$, you take a conic of $\mathbb{P}^2$ not passing through the points. It is equivalent to $C=2L$. You can thus see that no quadric of $\mathbb{P}^3$ contains it, because $C$ it is not equivalent to $-2K\_X$. There is a cubic $X'$ which contains $C$ is and only if $-3K\_X-C=7L-3E\_1-\dots-3E\_6$ is effective. Intersecting this system with the line $2L-E\_1-\dots-E\_6+E\_i$, we obtain $-1$, so the system should contain the $6$ lines, which is not possible. Hence, the cubic $X$ is the unique one which contains $C$. However, there are quartics containing $C$, because $-4K\_X-C=10L-4E\_1-\dots-4E\_6$ is effective. This can be seen by taking the union of two general quintics of $\mathbb{P}^2$ with $6$ double points. These curves are other rational sextics of $\mathbb{P}^3$. The system $-4K\_X-C$ being moreover base-point free, the ideal of $C$ in $\mathbb{P}^3$ is generated by the cubic and equations of degree $4$. You can do this easily in coordinates for a given curve, but this takes time.


$2)$ To obtain degree $6$ curves of genus $1$, you can take a cubic of $\mathbb{P}^2$ passing through three of the points, i.e. $C=3L-E\_1-E\_2-E\_3$. The same type of calculation gives you the ideal. 


You can of course continue with all curves of degree $6$. If you want more details on the type of the curves you obtain on the cubic surface, see "Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links" by S. Lamy and myself, especially Proposition 4.2. 


