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# Irreducibility of Equisingular Families of Curves

Thomas Keilen
Transactions of the American Mathematical Society
Vol. 355, No. 9 (Sep., 2003), pp. 3485-3512
Stable URL: http://www.jstor.org/stable/1194851
Page Count: 28
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## Abstract

In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface Σ it thus makes sense to look for conditions which ensure that the family V|D| irr(S1,... ,Sr) of irreducible curves in the linear system |D|l with precisely r singular points of types S1,... ,Sr is irreducible. Considering different surfaces, including general surfaces in P C 3 and products of curves, we produce a sufficient condition of the type $\sum_{i=1}^{r}\deg (X(S_{i}))^{2}<\gamma \cdot (D-K_{\Sigma})^{2}$, where γ is some constant and X(Si) some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.

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