How to find equations of $\mathbb{C}^*$-curves

Question

Fix positive integers $t_1,t_2,t_3$. Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by $$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{0\}\, \mbox{ sending } (\lambda, (x_1,x_2,x_3))\, \mbox{ to }\, (\lambda^{t_1}x_1,\lambda^{t_2}x_2,\lambda^{t_3}x_3).$$ Fix a point $(x_1,x_2,x_3) \in \mathbb{C}^3 \setminus \{0\}$. Denote by $C$ the closure of the orbit of the $\mathbb{C}^*$-action on the point $(x_1,x_2,x_3)$. I am trying to find the ideal $I$ of $C$ in $\mathbb{C}^3$. Of course, $I$ is contained in the ideal generated by $$(X_1/x_1)^{t_2}-(X_2/x_2)^{t_1}=0=(X_1/x_1)^{t_3}-(X_3/x_3)^{t_1}.$$ My question is: what are all the generators of $I$? Is it possible for $I$ to be generated by $2$ elements (not necessarily by the ones mentioned above)? Here we assume $x_1,x_2$ and $x_3$ are all non-zero.

EDIT If necessary assume that $t_1,t_2,t_3$ are pairwise coprime.

Answer 1

$I$ is generated by all binomials of the form $(X_1/x_1)^{a_1} (X_2/x_2)^{a_2} (X_3/x_3)^{a_3}-(X_1/x_1)^{b_1} (X_2/x_2)^{b_2} (X_3/x_3)^{b_3}$ where $a_1 t_1 + a_2 t_2 + a_3 t_3 = b_1 t_1 +b_2 t_2 + b_3 t_3$.

It suffices to consider all $6$-tuples $(a_1,a_2,a_3,b_1,b_2,b_3)$ which are not sums of two or more tuples of the same form and which satisfy $(a_1,a_2,a_3)\neq (b_1,b_2,b_3)$. These two conditions force $\min(a_1,b_1) =\min(a_2,b_2)=\min(a_3,b_3)=0$ so we must have one of the $a_i$ nonzero and two of the $b_i$ nonzero$ or the other way around.

For example if $a_1,b_2,b_3$ are nonzero then we must have $b_2,b_3<t_1$ since otherwise we could subtract off the tuple $(t_2,0,0,0,t_1,0)$ or $(t_3,0,0,0,0,t_1)$ as in your relations to write the tuple as a sum of two tuples of the same form. Symmetrical reasoning gives a bound for any element of the tuple. So these restrictions give a finite set of generators.

It will not always have two elements but certainly will sometimes, e.g. if $t_1=t_2=t_3=1$.