On the multiplicity of reducible relative stable morphisms

Abstract

Let $(Z,D)$ be a pair of a smooth surface and a smooth anti-canonical divisor. Denote by ${\mathfrak{M}}_{\beta }$ the moduli stack of genus $0$ relative stable morphisms of class $\beta$ with full tangency to the boundary. Let $C_1$ and $C_2$ be rational curves fully tangent to $D$ at the same point $P$ and assume that $C_1$ and $C_2$ are immersed and that $(C_1.C_2{)}_P=\min \{D.C_1,D.C_2\}$. Then we show that the contribution of $C_1\cup C_2$ to the virtual count of ${\mathfrak{M}}_{[C_1]+[C_2]}$ is $\min \{D.C_1,D.C_2\}$. As an example, we describe genus $0$ relative stable morphisms to $({\mathbb{P}}^2,(cubic))$ of degree $4$ with full tangency, and examine how they contribute to the relative Gromov-Witten invariant.

27 pages

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