# Automorphisms of a neighborhood of a negative curve

Let $$X$$ e a smooth complex surface and let $$C\subset X$$ be a smooth rational curve with negative self intersection.

Is there any known description of the automorphisms of a infinitesimal neighborhood of $$C$$ in $$X$$? Note that it is the same as a infinitesimal neighborhood of the zero section of the normal bundle of $$C$$, by Grauert's theorem.

• Please define "germ of a manifold $(X,C)$". If $X$ is birational to an Abelian surface $Y$, for example, then every automorphism of any dense Zariski open subset of $X$ arises from a unique biregular automorphism of $Y$. So there is no hope of classifying automorphisms only from the information of $C$ and the degree of the normal bundle of $C$ in $X$. However, if you intend "germ" to mean the formal completion of $X$ along $C$, that is quite different. – Jason Starr Jul 10 at 10:44
• Thanks Jason, I edited te question. I hope it is clearer. – Alan Muniz Jul 10 at 10:54