Principality of curves on sandwiched singularities

Given a surface $X$ obtained by blowing up a complete $\m$-primary ideal in the local ring of a point on a non-singular surface $S$, we determine the Picard group of $X$ and the divisor class groups of its singularities. Given a curve $C$ on $X$, we obtain various criteria for $C$ to be locally principal at these singularities. Our criteria are stated in terms of the projection of $C$ onto $S$. A minimal system of local generators of the defining ideal of $C$ is produced, as well as a formula for their number.

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Published 1 January 2010