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A Riemann singularity theorem for integral curves
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 5, October 2012
- pp. 1143-1165
- 10.1353/ajm.2012.0038
- Article
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We prove results generalizing the classical Riemann Singularity Theorem to the case of integral, singular curves. The main result is a computation of the multiplicity of the theta divisor of an integral, nodal curve at an arbitrary point. We also suggest a general formula for the multiplicity of the theta divisor of a singular, integral curve at a point and present some evidence that this formula should hold. Our results give a partial answer to a question posed by Lucia Caporaso in a recent paper.