# 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.