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.


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. 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. 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.


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