Contact Order Of A Stable Log Curve At Marks And Nodes

definition

Let \((C/S, \mathbf p, f)\) be a stable log map with target \(X/B\), and assume that \(S = \Spec (Q'\to k)\) is a log point with \(Q'\) some arbitrary sharp fs monoid and \(k\) an algebraically closed field. There are two log morphisms of interest on \(C\) which we will use to define contact order at nodes \(q\in C\) and marks \(p\in C\): these two morphisms are \(f:C\to X\) and \(\pi:C\to S\), which induce maps on ghost sheaves

  • \(\psi := \overline \pi^\flat :\pi^{-1}Q' \to \overline{\mathcal M}_C \)
  • \(\varphi := \overline{f}^\flat:f^{-1}\overline{\mathcal M}_X\to \overline{\mathcal M}_{C}. \)

Structure of \(\psi\)

The map \(\psi\) is an isomorphism away from the marks and nodes of \(C\), a fact which should follow from the classification of log structures on curves. The sheaf \(\overline{\mathcal M}_C\) has stalks \(Q'\oplus \mathbb N\) and \(Q'\oplus_{\mathbb N}\mathbb N^2\) at marks and nodes respectively. The latter monoid is a fibered sum of monoids determined by the diagonal map \(\mathbb N\to \mathbb N^2\) defined \(1\mapsto (1,1)\) and \(\mathbb N\to Q'\) defined \(1\mapsto \rho_q\). See this note for a discussion of the nodes.

  • At marks, \(\psi_p\) is the inclusion \(Q'\to Q'\oplus \mathbb N\).
  • At nodes, \(\psi_q\) is the inclusion \(Q' \to Q'\oplus_{\mathbb N}\mathbb N^2\).

Structure of \(\varphi\)

This is slightly more difficult. Set \(P_x = f^*\overline{\mathcal M}_{C,x} = \overline{\mathcal M}_{X, \underline f(\overline x)}\) so that at a stalk \(\varphi\) is given by \(\varphi_{\overline x}: P_x\to \overline{\mathcal M}_{C,\overline x}\). We then have three cases:

  1. \(x = \eta\) is a generic point, then we get a local homomorphism of monoids
\begin{align*} \varphi_{\overline \eta}:P_\eta \to Q'. \end{align*}
  1. \(x = p\) is a marked point, then we have a map
\begin{align*} \varphi_{\overline p}:P_p\to Q'\oplus \mathbb N \end{align*}
  1. \(x = q\) is a node, then we have a morphism
\begin{align*} \varphi_{\overline q}:P_q \to Q'\oplus_{\mathbb N} \mathbb N^2. \end{align*}

Fibered sums can be bad in general, but because everything is integral we’re alright. We can now define contact order.

Definition

Let \((C/S, \mathbf p, f)\) be a stable log map with target \(X\), as above. Let \(p \in C\) be a mark

  • Let \(p\in C\) be a marked point. The contact order at \(p\) is \(u_p\in P^\vee_p\) defined
\begin{align*} u_p:P_p\xrightarrow{\varphi_{\overline p}} Q'\oplus \mathbb N\xrightarrow{\text{pr}_2}\mathbb N. \end{align*}
  • Let \(q\in C\) be a node contained in the closures of both \(\eta_1\) and \(\eta_2\). Furthermore let \(\chi_i:P_q\to P_{\eta_i}\) be the generization maps. The contact order at \(q\) is the homomorphism \(u_q:P_q\to \mathbb Z\) uniquely defined by the equality
\begin{align*} \varphi_{\overline \eta_2}(\chi_2(m)) - \varphi_{\overline \eta_1}(\chi_1(m)) = u_q(m) \cdot \rho_q, \end{align*}

where \(\rho_q\neq 0\) is the image of \(1\) in \(Q'\) given by the map \(\mathbb N \to Q'\) defining the structure of \(\overline{\mathcal M}_{C,q} \cong Q'\oplus_{\mathbb N} \mathbb N^2\).

I think the contact order at the node should be thought of as the contact order/order of tangency along one component minus the contact order/order of tangency along the other component, where the difference is ordered with respect to the orientation chosen at the node. Here’s a more detailed discussion.

When two components \(C_1\) and \(C_2\) of \(C\) with generic points \(\eta_1\) and \(\eta_2\) meet at a node \(q\in C\) we obtain the following diagram:

The morphisms \(\varphi_x\) are discussed above, \(\chi_i\) are generization maps, and \(\iota\) is the embedding of \(Q\oplus_\mathbb N\mathbb N^2 \) into \(Q\times Q\) discussed here. Commutativity of this diagram implies that \(\varphi_q\) is entirely determined by the two \(\varphi_{\eta_i}\) maps. Since \((m_1,m_2)\in Q\times Q\) is in the image of \(\iota\) precisely when there is some \(n\in \mathbb Z\) so that \(m_1 - m_2 = n\cdot \rho_q\), there is a homomorphism \(u_q:P_q\to \mathbb Z\) defined by \[u_q(m) = \frac{\varphi_{\eta_1}(\chi_1(m)) - \varphi_{\eta_2}(\chi_2(m))}{\rho_q}.\] This is the contact order at the node.

Standard log point In fact, if we further restrict to the case of the standard log point, so that \(Q = \mathbb N\), then the maps \(\varphi_\eta:P_\eta\to \mathbb N\) can be thought of as elements \(V_\eta \in P_\eta^\vee\). Dualizing the generization maps gives inclusions \(\iota_{q,\eta}:P^\vee_\eta\to P^\vee_q\), and \(u_q\in (P_q^{gp})^\vee\) already, so we can state the contact order equation at a node \(q\) in this situation instead as \[u_q\cdot e_q = \iota_{q,\chi_1}(V_{\eta_1}) - \iota_{q,\eta_2}(V_{\eta_2}).\] Here we write \(e_q\) instead of \(\rho_q\) to emphasize it is just a number in \(\mathbb N\setminus \{0\}\).

NOTE: The definition of the contact order of the node requires a choice of orientation at the node. Such a choice can be implemented by choosing an orientation of the corresponding edge in the dual intersection graph of the underlying pre-stable curve.

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