Here we study the moduli space of stable log maps \(LM(X, \beta)\)with target \(X = \mathbb P^2\) and logarithmic curve class \(\beta\) consisting of
- ordinary curve class \(A = [\ell] \in A_1(X)\) where \(\ell\) is a line
- genus map 0: \(g = 0\)
- marked points \(n\) left undetermined, we’ll consider many cases
- contact order \(p_1,...,p_n\) also undetermined.
The first thing we do is consider the moduli space of ordinary stable maps.
Ordinary Stable Maps
\(X = \mathbb P^2\) with \(g = 0, d = 1\)
First, the case of no marked points:
\(n = 0\)
We’re looking for degree 1 stable maps to \(\mathbb P^2\) up to isomorphism. Each degree \(1\) map is the parameterization of a line, and by identifying stable curves we ignore parameterizations. Hence the image curve \(C\) of a class \([C, f]\in \overline{M}_{0,0}(\mathbb P^2, 1)\) determines the class; hence \(\overline M_{0,0}(\mathbb P^2, 1) = \Gr(1, \mathbb P^2) = \mathbb P^{2*}\).