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Let $C$ be a nodal curve over an algebraically closed field $k$. A deformation of $C$ over an Artinian ring $A$ over $k$ consists of a flat scheme $C'$ over $A$ and a closed immersion $i: C \to C'$ s.t. $i \times_{Spec\ A} Spec\ k: C \to C' \times Spec\ k$ is an isomorphism.

The dual graph $G$ of $C$ is a graph with irreducible components of $C$ as vertices and nodes of $C$ as edges.

Reading a paper I came across the notion of a first order deformation of $C$ preserving $G$ ("first order" means that $A = k[\varepsilon]/(\varepsilon^2))$ . What does "preserving $G$" mean? Unfortunately, I could not find any definition or explanation on the web or in the literature.

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I've heard of that duality but I am not sure at the moment. Maybe you are reading about (pointed) stable curves? Maybe this link will help: projectivepress.com/moduli/moristablecurves.pdf –  math-visitor Jun 8 '12 at 11:26
    
I am reading about maps of pointed stable curves, particularly Fultons and Pandharipandes "Notes on stable maps and quantum cohomology". –  finite Jun 9 '12 at 8:47

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