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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.

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