ag.algebraic geometry – Gromov-Witten invariants of cocharacter closures in toric varieties


$require{AMScd}$

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
$$mathbb{C}^times xrightarrow{h} (mathbb{C}^times)^n xrightarrow{iota} X qquad text{where}qquad htext{ is a group homomorphism}$$

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, a cocharacters are abundant since a cocharacter $u$ is equivalent to an element of $mathbb{Z}^n$ via the map
$$a = (a_1,dots,a_n) mapsto u_a qquadtext{with}qquad u_a(z) = (z^{a_1},dots,z^{a_n})$$
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.