X(V, M) = Dim Lm + 1 LEMMA 1. a Specialization of The
34 MA THEMA TICS: J. IGUSA PROC. N. A. S. 4R.Bellman, Dynamic Programming and ContinuousProcesses (RAND Monograph R-271, 1954). 5R. Bellman, "Dynamic Programming and a New Formalism in the Calculus of Variations," these PROCEEDINGS, 40, 231-235, 1954. 6 R. Bellman, "Monotone Convergence in Dynamic Programming and the Calculus of Vari- ations," ibid., (these PROCEEDINGS, 40, 1073-1075, 1954). 7 E. Hille, Functional Analysis and Semi-groups ("American Mathematical Society Colloquium Publications," Vol. XXXI [1948]). 8 Cf. ibid., p. 71. 9 Ibid., p. 388. 10 V. Volterra, Leqons sur les fonctions des lignes (Paris: Gauthier-Villars, 1913). ARITHMETIC GENERA OF NORMAL VARIETIES IN AN ALGEBRAIC FAMILY* BY JUN-ICHI IGUSA DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, AND KYOTO UNIVERSITY, JAPAN Communicated by Oscar Zariski, October 28, 1954 It is well known that linear equivalence of divisors on a fixed ambient variety is preserved by specialization.' In this paper we shall show that the above assertion remains valid even when the ambient variety varies under specialization. This fact will be used as a lemma in our later paper. Here we shall derive the following theorem as an immediate consequence. If a normal variety V' is a specialization of a positive cycle V, then V is also a normal variety, and they have the same arith- metic genus. In the case of curves this assertion was proved by Chow and Nakai,2 and in our proof we shall use some of their ideas. We note also that a slightly less general result was proved in the classical case by Spencer and Kodaira quite recently.3 Let yr be a variety of dimension r in a projective space L'.
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