Identifying variable points on a smooth curve
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LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x 1, …,x n ] an unorderedn-tuple of not necessarily distinct points onX. Byf x :X →Y x we denote the normalization which identifies thex 1, …,x n and maps them to the only and universal singularity of a complex curveY x . Thenf x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ X of all complex-analytic quotients ofX and construct a morphismS n (X) →Q X such that each and everyf x corresponds to the image of the pointx on then-fold symmetric powerS n (X). For everyn ≥ 2 the mappingS n (X) →Q X is a closed embedding; the points of its image have embedding dimensionn(n − 1) inQ X . HenceS 2(X) is a smooth connected component ofQ X . On the other hand, a deformation argument yields thatS n (X) is part of the singular locus of the complex spaceQ X provided thatn ≥ 3.
1991 Mathematics Subject ClassificationPrimary 32G13 Secondary 32S30 32S45 14H15 14H20
Key words and phrasesIdentifying points complex curves parameter spaces normalizations universal singularities moduli of quotients
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- Schuster, P. M.:Moduln singulärer Kurven mit vorgegebener Normalisierung. Diss., Univ. München 1995. Also at Verlag Mainz, Aachen 1996Google Scholar