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Transactions of the American Mathematical Society

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On specializations of curves. I

Author: A. Nobile
Journal: Trans. Amer. Math. Soc. 282 (1984), 739-748
MSC: Primary 14H10; Secondary 14H20
MathSciNet review: 732116
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Abstract: The following is proved: Given a family of projective reduced curves $ X \to T$ ($ T$ irreducible), if $ {X_t}$ (the general curve) is integral and $ {X_0}$ is a special curve (having irreducible components $ {X_1}, \ldots ,{X_r}$), then $ \sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})} $, where $ g(Z) = $ geometric genus of $ Z$. Conversely, if $ A$ is a reduced plane projective curve, of degree $ n$ with irreducible components $ {X_1}, \ldots ,{X_r}$, and $ g$ satisfies $ \sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac{1} {2}(n - 1)(n - 2)} $, then a family of plane curves $ X \to T$ (with $ T$ integral) exists, where for some $ {t_0} \in T,{X_{{t_0}}} = Z$ and for $ t$ generic, $ {X_t}$ is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.

References [Enhancements On Off] (What's this?)

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Keywords: Family of curves, genus, nodal curve, assigned singularities
Article copyright: © Copyright 1984 American Mathematical Society

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