I expect that the fixed locus of the moduli space of stable log maps should exactly be those ordinary stable maps which lift to log maps. We have conditions coming from two sources:
- conditions on the ordinary fixed locus
- conditions for an ordinary stable map to lift to a stable log map
Lemma
Let \((C/S, f, p)\) be a stable log map over the standard log point \(S = \Spec (\mathbb N\to k)\) to \((X, D)\) a toric variety with divisorial log structure given by \(D\), where \(D\) is a divisor whose support is contained in the toric divisor of \(X\). Let \(\Gamma\) be the dual graph of \(C\). If \(C\) is a fixed point of \(\mathcal M(X, \beta)\) the moduli space of stable log maps, then the following conditions hold:
- Each vertex of \(\Gamma\) must be mapped to either a full dimensional cone of \(\Sigma(X)\) (component contracted to a fixed point) or to a codimension 1 cone of \(\Sigma(X)\) (covers a line in \(X\)).
- If \(C_0\subset C\) is not contracted, then every point of \(C_0\) mapped to a fixed point of \(X\) must be a marked point or a node. In particular, every ramification point must be a marked point or a node. In terms of discrete data, edges and legs attached to the vertex \(v_0\in \Gamma\) associated to \(C_0\) must not be contained in a proper face.