Two-marked logarithmic Gromov-Witten invariant of a surface pair

Let $X$ be a smooth projective surface over $\mathbb{C}$, let $D\subset X$ be a smooth anticanonical divisor, and let $\beta \in \mathrm{NE}(X)$ be an effective curve class with $d=\beta \cdot D\ge 2$. For positive integers $p,q$ with $p+q=d$, suppose the source's proper moduli stack of basic genus-zero stable logarithmic maps to the divisorial log pair $(X,D)$, with two marked points of contact orders $p$ and $q$, is equipped with its source virtual class. Write it as ${\mathcal{M}}_{0,(p,q)}((X,D),\beta )$ and write ${\mathrm{ev}}_1$ for evaluation at the first marked point, whose image lies in $D$.

For a point class $[x]\in H^2(D;\mathbb{Q})$, the source's two-marked logarithmic Gromov-Witten invariant is $R_{0,(p,q)}((X,D),\beta )={\int }_{[{\mathcal{M}}_{0,(p,q)}((X,D),\beta ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}{\mathrm{ev}}_1^{\ast }[x]\mathrm{.}$ The source convention fixes the first contact to $p$ at the constrained point and the second contact to $q$ at an unconstrained marked point. The invariant is a rational virtual degree, not an enumerative count unless a separate transversality and multiplicity argument proves that interpretation.

For disconnected $D={\coprod }_a{D}_a$, the two boundary markings carry component indices as discrete data. A component-assigned stack records contact $p$ on $D_a$ and $q$ on $D_b$; the stack without indices is the disjoint union of all compatible assignments, with virtual class their sum. The point class $[x]$ means an actual chosen point $x\in D_a$ on a specified component: its evaluation insertion kills assignments of the first marking to another component. Write $R_{0,(p,q)}^{a,b}$ for the assigned invariant. For this chosen $x\in D_a$, the index-free invariant sums over compatible $b$; changing $a$ can change the value. A sum of point classes over components instead sums these invariants and is a different insertion. For connected $D$ this reduces to the original convention.

This is a conditional definition: the source's proper logarithmic moduli construction and its virtual class are part of the input. The ordinary stable-map and Behrend-Fantechi definitions cited below do not by themselves construct that logarithmic moduli stack or its logarithmic obstruction theory.

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