$\require{AMScd}$
Let $X$ be a toric projective variety with dense algebraic torus $\iota:(\mathbb{C}^\times)^n \to X$, and let $u:\mathbb{C}^\times \to X$ be a cocharacter, by which I mean a map admitting a factorization of the form $$\mathbb{C}^\times \xrightarrow{h} (\mathbb{C}^\times)^n \xrightarrow{\iota} X \qquad \text{where}\qquad h\text{ is a group homomorphism}$$
Definition. The closure $\bar{u}:C \to X$ of the cocharacter $u$ is the unique extension of $u$ to a singular toric curve $C$ that commutes with the $\mathbb{C}^\times$-action on $\mathbb{C}^\times$ and $C$.
This construction seems pretty natural to me. Furthermore, cocharacters are abundant since a cocharacter $u$ is equivalent to an element of $\mathbb{Z}^n$ via the map $$a = (a_1,\dots,a_n) \mapsto u_a \qquad\text{with}\qquad u_a(z) = (z^{a_1},\dots,z^{a_n})$$ However, I am having trouble finding information about these curves. For example, I am interested in the following question.
Question 1. Are there other characterizations of the curves arising from this construction?
I am also interested in the Gromov-Witten theory of these curves. All that I can ask here is the following vague question.
Question 2. Is there some sense in which the curves $\bar{u}_a$ has a "non-trivial count in Gromov-Witten theory"?
I'm lookin for an answer like: for each $a \in \mathbb{Z}^n$, there exists a $0$-dimensional moduli space of stable curves $\overline{\mathcal{M}}_{g,n}(X,A)$ that naturally includes $\bar{u}_a$ (somehow) and where $GW^{X,A}_{g,n} \neq 0 \in H_0(X)$. This is almost certainly too specific, but anything in this general direction would be great.
