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. 



The closure of a locus in $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$
It turns out that the answer is yes and these conditions are sufficient.


This question was answered in greater generality by Gathmann ([Absolute and relative Gromov-Witten invariants of very ample hypersurfaces.](https://arxiv.org/abs/math/9908054) *Duke
Math. J.*, 2002, Proposition 1.14) building off of earlier work of Vakil ([The enumerative geometry of rational and elliptic curves in projective space.](http://math.stanford.edu/%7Evakil/files/re4.pdf) *J. Reine Angew. Math.*, 2000, Theorems 4.13 and 6.1).


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](https://arxiv.org/pdf/2108.05324.pdf).


