Balancing Condition For Stable Log Curves

See

• [GSa, Proposition 1.15]

The balancing condition is a slight generalization of the balancing condition in tropical geometry that makes sense in the setting of stable log curves over the standard log point when we consider associated maps of tropical curves.

Let \((C/S, f, p)\) be a stable log map to \(X\) with \(S = \Spec(\mathbb N\to k)\) the standard log point. The balancing condition will be a statement that holds at a vertex of the dual graph of \(C\), so fix an irreducible component \(g:D\hookrightarrow C\) with generic point \(\eta \in D\). Further assume \(D\) is smooth (the only way it isn’t smooth is if \(D\) has a self-node, in which case we replace it with its normalization \(\widetilde D\)). The stalk of the ghost sheaf at a node \(q \in C \) is then \(\overline{\mathcal M}_{C,q} = S_{\rho_q}\), where \(S_{\rho_q} \subseteq \mathbb N^2\) is the submonoid generated by \((\rho_q,0), (0,\rho_q) \) and \( (1,1) \), and we can choose the generization map \(\chi_q:\overline{\mathcal M}_{C,q} = S_{\rho_q}\to \overline{\mathcal M}_{C, \eta} = \mathbb N\) to be projection onto the second coordinate (this is a choice of orientation at the node) so that \(\chi_q(a,b) = b \). Additionally denote \(P_x = \overline{\mathcal M}_x\) for any point \(x \in C\). Below we’ll use \(Q\) to denote the factor of \(\mathbb N\) coming from the standard log point to differentiate from the other \(\mathbb N\) factors coming from Kato’s theorem.

To define it we need two sets of maps, the contact order maps and the \(\tau\) maps which don’t have names.

Contact orders At marked points \(p\) and nodes \(q\) we obtain maps on stalks \(\overline{\mathcal M}_x\to \overline{\mathcal M}_{C,x}\). At marked points \(p\in C\) this map is \(P_p\to \mathbb N\oplus Q\) and at nodes \(q\in C\) it is \(P_q\to S_{\rho_q}\), using Kato’s theorem on the structure of log smooth curves. See also the theorem describing pushouts of integral monoids. The contact orders at marks and nodes are the compositions of these morphisms with maps to \(\mathbb N\) and \(\mathbb Z\) respectively: \[u_p:P_p \xrightarrow{\varphi_p} \mathbb N\oplus Q\xrightarrow{\text{proj}_1} \mathbb N\] and \[u_q:P_q\xrightarrow{\varphi_q} \mathbb N^2\oplus_{\mathbb N} Q \to \mathbb Z. \]

The other two maps The other two maps are obtained as follows. Let \(\eta\) be the generic point of a component \(g:D\hookrightarrow C \), and

Each global section of \(\overline{\mathcal M}\) and \(\overline{\mathcal M}_C\) determines a line bundle on \(C\), which we can restrict to \(D\) and then take the degree to obtain a map to \(\mathbb Z\). This gives us the maps \[\tau^X_\eta:\Gamma(D, g^*\overline{\mathcal M})\to \Pic(D)\xrightarrow{\deg}\mathbb Z\] \[\tau_\eta^C:\Gamma(D, g^*\overline{\mathcal M}_C)\to \Pic(D)\xrightarrow{\deg}\mathbb Z.\]

The map \(\tau^X_\eta\) is entirely determined by \(\mathcal M\) and \(f\), we have no control over it. The only thing we can say in general is that it is the \(0\) map when \(D\) is contracted to a point in \(X\).

The map \(\tau_\eta^C\) is similarly determined by \(g^*\mathcal M_C\), but there is more we can say about it, namely, we can write its domain explicitly in terms of nodes \(q \in D\) and marks \(p \in D\).

In particular, if \(((a_q,b_q),n_p)\) represents a global section of \(g^*\overline{\mathcal M}_C\) then all the \(b_q\)’s match. It is then a lemma that

Balancing condition The way we relate the contact orders to the \(\tau\) maps is via the following diagram:

It commutes because, according to the last sentence of page 462 in Log GW, “\(f^\flat\) must induces isomorphisms of torsors.” The formula \(\tau^X_\eta = \tau^C_\eta\circ \varphi\) is an equation in the dual group \(\Hom(\Gamma(D, g^*\overline{\mathcal M}^{gp}), \mathbb Z)\), which we denote \(N_D\). It’s the inductive limit of morphisms \(\iota_{D, x}: (P^\vee_{\eta})^{gp}\to (P^\vee_x)^{gp}\) given by dualizing the generization maps for \(x\in D\). These maps are all isomorphisms except at special points, so if we let \(\Sigma \subset D\) be the collection of marks and nodes, then \[N_D = \Gamma(D, g^*\overline{\mathcal M}^{gp})^\vee = \lim_{x\in D}(P^\vee_x)^{gp} = \bigoplus_{x\in \Sigma}(P^\vee_x)^{gp}\Big/\sim\] where \((0,...,0,\iota_{D, x}(a), 0,...0)\sim (0,....,0,\iota_{D,x'}(a), 0,...,0)\) for any \(x,x' \in \Sigma\) and \(a\in (P^\vee_\eta)^{gp}\). This means an element of \(N_D\) can be represented as a tuple \((a_x)_{x\in \Sigma}\) as long as we keep the relations in mind.

The morphisms \(u_p\) and \(u_q\) are also elements in \(N_D \), and thus the contact orders of marked points and nodes form a tuple \((u_x)_{x\in \Sigma}\in N_D\). Similarly, \(\tau^X_\eta\) corresponds to a tuple \((\tau_x)_{x\in \Sigma}\in N_D\). The balancing condition is then simply

which is Proposition 1.15 in Log GW. The proof involves showing that \(\tau_{\eta}^C\circ \varphi\) is literally \((-u_x)_{x\in \Sigma}\), and then applying the commutativity of the above diagram.