# The closure of a locus in $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$

Consider the closure $$K \subset \overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$$ in the stable maps space of the locus $$K_0$$ of maps $$(f : C \to \mathbb{P}^1, p_1, \ldots, p_n)$$ where $$C \cong \mathbb{P}^1$$ is smooth and the set of marked points is exactly the preimage of $$\infty \in \mathbb{P}^1$$; $$\{p_1, \ldots, p_n\} = f^{-1}(\infty)$$. I want to understand the boundary strata $$K \setminus K_0$$.

There is an obvious necessary condition for a pointed stable map $$(f : C \to \mathbb{P}^1, p_1, \ldots, p_n)$$ to be in $$K$$, which is that $$f(p_i) = \infty$$ for each $$i$$. In particular, $$K$$ is contained in the intersection of the evaluation loci $$ev_i^{-1}(\infty)$$ for $$i = 1, \ldots, n$$ where $$ev_i : \overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n) \to \mathbb{P}^1$$ are the usual evaluation maps. Let us call this Condition 0.

However, this condition is not sufficient. For example there is a stratum paramerizing $$f : C_0 \cup_p C_1 \to \mathbb{P}^1$$ where the restriction $$f_0 : C_0 \to \mathbb{P}^1$$ is a degree $$n$$ map from a smooth rational curve with $$f_0(p) = \infty$$, $$f$$ constant on $$C_1$$, and all the marked points lying on $$C_1$$. This stratum is contained $$ev_i^{-1}(\infty)$$ for each $$i$$ but is larger dimensional than $$K$$.

The reason is that being in the closure of $$K_0$$ imposes some extra conditions on the map. For degree reasons, a map $$f : C \to \mathbb{P}^1$$ in $$K_0$$ must be unramified at all the marked points $$p_i$$ but the only way marked points can collide is if the map becomes ramified at $$\infty$$. This gives us a condition on certain nodes lying over $$\infty$$ which I think can be phrased as follows:

Condition 1: Let $$C_0 \subset C$$ is a connected union of components lying in $$f^{-1}(\infty)$$. Suppose $$C_0$$ is attached to components $$C_1, \ldots, C_k$$ at points $$x_1, \ldots, x_k$$ where $$f|_{C_i}$$ is non-constant. Then the number of marked points lying on $$C_0$$ is the sum of the ramification of $$f|_{C_i}$$ at $$x_i$$.

Condition 2: For each component $$C_0 \subset C$$ with $$f|_{C_0}$$ non-constant, each point of $$f|_{C_0}^{-1}(\infty)$$ is either a marked point or a node of $$C$$.

These two conditions are related but its not immediately clear to me exactly how. Note that a general point of the stratum described above not contained in $$K$$ satisfies neither condition $$1$$ nor $$2$$.

Question 1: Are conditions $$0$$, $$1$$ and $$2$$ sufficient for a stable map to be contained in the closure of $$K$$? If not, is there some other description of the boundary strata of $$K$$?

I think a quick dimension count shows that the dimension of the strata satisfying conditions $$0$$, $$1$$ and $$2$$ have dimension at most the dimension of $$K$$, and in fact strictly smaller than the dimension of $$K$$ if you exclude $$K_0$$ itself. This gives some evidence that Question 1 has a positive answer. For example, if one knew that the expected dimension of the locus satisfying these conditions is the dimension of $$K$$, then these dimension bounds would be enough. However, I don't know how to show this.

We wrote a direct proof of the special case considered in the question (namely target $$(\mathbb{P}^1, \infty)$$) in Section 2 of this preprint.