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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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?)

  • [1] G. Albanese, Sulle condizioni perchè una curva algebraica riducible si possa considerare come limite di una curva irreducibile, Rend. Circ. Mat. Palermo (2) 52 (1928), 105-150.
  • [2] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 0262240 (41 #6850)
  • [3] W. Fulton, Algebraic curves, Benjamin, New York, 1968.
  • [4] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [5] Heisuke Hironaka, On the arithmetic genera and the effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 30 (1957), 177–195. MR 0090850 (19,881b)
  • [6] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911 (42 #1813)
  • [7] Augusto Nobile, On families of singular plane projective curves, Ann. Mat. Pura Appl. (4) 138 (1984), 341–378. MR 779551 (86f:14015), http://dx.doi.org/10.1007/BF01762552
  • [8] F. Severi, Vorlesungen über Algebraische Geometrie, Teubner, Leipzig, 1921.
  • [9] B. Tessier, Resolution simultané. I, II, Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., Vol. 777, Springer-Verlag, Berlin and New York, 1980.
  • [10] Oscar Zariski, Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order 𝑛 and with a given number 𝑑 of nodes, Amer. J. Math. 104 (1982), no. 1, 209–226. MR 648487 (83m:14044), http://dx.doi.org/10.2307/2374074
  • [11] O. Zariski, Contributions to the problem of equisingularity, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 261–343. MR 0276240 (43 #1987)
  • [12] -, Algebraic surfaces, 2nd ed., Springer-Verlag, Heidelberg, 1971.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0732116-X
PII: S 0002-9947(1984)0732116-X
Keywords: Family of curves, genus, nodal curve, assigned singularities
Article copyright: © Copyright 1984 American Mathematical Society