When does an ordinary stable map \((C/S, f, p)\) lift to a stable log map? We’ll go through some criterion.
Suppose the target \(X\) has divisorial log structure given by a normal crossings divisor \(D\). We’ll put a fine saturated log structure \(\mathcal M_C\) on \(C\) and assume \(S\) is the standard log point. We then ask, is \((C/S, f, p)\) a stable log map?
Observation 1: Generically only marks and nodes can be mapped to the divisor
The title isn’t quite right: what is actually true is that \(f^ *\overline{\mathcal M}\) can only jump rank at the nodes and marks.
Proof
Suppose \(\eta \in C\) is the generic point of some component \(D \subset C\), and let \(x\in D\) be a closed point which is neither a node or a mark. We then have the following diagram
where the horizontal arrows are generizations. The generization map \(\chi_X \) is surjective, and since \(\overline{ \mathcal M}_X\) is the ghost sheaf of a fine log structure, it is obtained by localizing along a face of \(\overline{ \mathcal M}_{X,x}\) and then quotienting out by the invertible elements. Hence if it not an isomorphism, then there must be some \(0\neq m\in \overline{f^*\mathcal M}_{X,x}\) such that \(\chi_X(m) = 0 \in f^*\overline{\mathcal M}_{X,\eta}\). Since \(x\in C\) is not a special point, the generization map \(\chi\) is an isomorphism (see Kato’s theorem). Commutativity of the above diagram therefore gives us that \[\overline{f}^\flat_x(m) = 0.\] This contradicts locality because toric monoids are sharp, hence \(\chi_X\) must be an isomorphism.
In particular, \(f^*\overline{\mathcal M}_{X,x}\) can only jump rank at marked points and nodes.
For a generic stable map \((C/S, f, p)\) there should be finitely many points in \(f^{-1}(D)\), and since \(f^*\overline{\mathcal M}\) necessarily jumps at all of them each intersection point of \(f(C) \cap D\) must occur at the image of a node or a mark. This does not account for the scenario in which \(f(C) \subset D\), here jumps in the rank of \(f^*\overline{\mathcal M}\) indicate points where \(C\) falls into higher codimensional strata.
Observation 2: The degree of the stable map constrains the possible choices for contact orders of the marked points.
The degree of \(f\) and the contact orders of the marked points are related via the following formula:
This is a corollary of the formula relating degree and contact orders of marks and nodes on a log stable curve together with the balancing condition.
Corollary
Let \((C/W, \mathbf p, f)\) be a log stable curve with \(W\) a log point. Let \(\mathcal L_s\) denote the line bundle associated to a global section \(s\in \Gamma(X, \overline{\mathcal M}_X)\) and let \(u_{p_i}\) denote the contact order of \(p_i\). Then
\begin{align*} \deg \underline f^*\mathcal L_s = -\sum_{i=1}^n\langle u_{p_i}, s\rangle. \end{align*}The point is that, when we don’t restrict to a single component of \(C\), the contact orders of the nodes cancel and we only get the contribution of the marks.
Formula For The Degree And Contact Orders Of Marks On A Log Stable Curve