A log or punctured log map over a log point \(\Spec (Q\to k)\) has a tropicalization, a map of cone complexes \(\Gamma \to \Sigma(X)\) over a cone \(\omega = Q^\vee_{\mathbb R}\). A type is what survives when you forget the continuous data of such a family. They appear as the indices in stratifications of moduli spaces of log maps.
Definition
(Class). The data \(\beta = (g, A, n, u_p)\) is called the class associated to a log map \((C, f, p_1,...,p_n)\) where
- \(g \) is the genus of the source curve
- \(A \) is the curve class \(f_*[C]\)
- \(n\) is the number of marked points
- \(u_{p_i}\) is the contact order at the marked point \(p_i\)
The moduli space \(M(X/B, \beta)\) is then the moduli space of stable maps of class \(\beta\). The data \(\underline \beta\) is precisely the combinatorial data specified in ordinary Gromov-Witten theory. See definitions 2.15-2.16 in the Decompositions paper. It is this moduli space whose degenerate fiber is decomposed in the main decomposition theorem.
The “type” of a log map can be retrieved before or after tropicalization. We have two slightly different definitions depending on whether the input is a log map or a tropical map.
Definition
(Type of a log map.) Suppose \((C, f, \mathbf p)\) is a log map over a log point \(S = \Spec(Q\to k)\). It is the quadruple \(\tau = (G, \mathbf g, \sigma, u)\) where
- \(G = V(G) \cup E(G) \cup L(G)\) is the dual intersection graph of a source curve where vertices \(V(G)\) are the generic points of components, edges \(E(G)\) are the nodes and legs \(L(G)\) are the marked points
- \(\mathbf g:V(G) \to \mathbb N\) is the genus of a vertex
- \(\sigma:V(G)\cup E(G) \cup L(G) \to \Sigma(X)\) is the “cone function” which assigns to a point \(x\) the cone \(\sigma(x) = \left(\overline{\mathcal M}_{X, f(x)}\right)^\vee_{\mathbb R}\) in the tropicalization of \(X\), it is the piece of \(X\) to which \(x\) maps in the tropicalization
- \(u = \{u_p, u_q\}\) is the contact order of the marks and the nodes.
Attached to this construction is the basic monoid \(Q(\tau)\), which depends only \(G\), \(\sigma\) and \(u\). We emphasize that this is discrete data associated to a log map.
Definition
(Type of a family of tropical maps.) The type of a family of tropical maps is the same data as the type of a stable log map, but read off from the tropical data.
Geometric ~
\(,\quad\) Tropical ~
Start with a family of tropical curves \(\Gamma \to \omega\) over a cone \(\omega\) together with a tropical map \(h:\Gamma \to \Sigma(X)\) from \(\Gamma\) to a cone complex \(\Sigma(X)\). We think of \(\Gamma\) here as a genus-decorated metric graph \((G, \mathbf g, \ell)\) rather than the cone complex associated to it. The type is then
\begin{align*} \tau = (G, \mathbf g, \sigma, \mathbf u) \end{align*}where
- \(G\) is the underlying graph
- \(\mathbf g:V(G)\to \mathbb N\) is the genus decorating function
- \(\sigma:V(G)\cup E(G) \cup L(G) \to \Sigma(X)\) is the map assigning each piece of graph data to the cone in which its image lands
- \(\mathbf u\) is the contact orders where \(u_E \in N_{\sigma(E)}\) is defined by \(h(v') - h(v) = \ell(E) \cdot u_E\) along each edge and \(u_L\in N_{\sigma(L)}\) is the outgoing slope of each leg.
Notice that the length \(\ell\) appearing as part of the data of \(\Gamma \)is forgotten.
Compared the the type of a stable log map, now \(\sigma(x)\) is the minimal cone containing the image of \(h(\omega_x)\) and the contact orders are the images of tangent vectors \((0,1)\) under \(h\) (which makes them much more interpretable in my opinion).
Type Of A Family Of Tropical Maps Defn
The equivalence of “type of a log map” and “type of a family of tropical maps” justifies the convention of simply calling both of these “type”.
Definition
(Decorated tropical type.) It’s the same as the notions of type but with the extra decoration of a curve class: we add a map \(A:V(G)\to H^+_2(X)\) mapping components of \(C\) i.e. vertices in \(G\) to curve classes, and we denote the total class \(|A| = \sum_{v\in V(G)} A(v)\). The decorated type is denoted
\begin{align*} \tilde \tau = (\tau, A). \end{align*}This is the level at which the decomposition formula functions; two rigid tropical maps with the same underlying types but with different curve classes are counted separately.
Here are some peripheral notions in the theory.
Contraction morphisms: The morphisms of types are contractions \(\tau\to \tau'\) given by contracting an edge. Their geometric orign/analogue is generization maps; contracting an edge is like smoothing a node i.e. moving into a more generic strata. This is what “the type jump at boundary strata” refers to. See Lemma 2.30 in the Decompositions paper.
Rigid tropical types: We say a type \(\tau = (G, \mathbf g, \sigma, u)\) is rigid if its basic monoid is \(\mathbb N\).