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 arithmetic 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'. The Hilbert characteristic function X(V, t) of V' is then a polynomial in t of degree r. We shall denote its constant term by x(V), and we call it the arithmetic genus of V. The usual arithmetic genus pa(V) of V is related to x(V) by x(V) = 1 + (- 1) pa(V). If V' has no multiple subvariety of dimension r - 1, we can consider linear systems on Vr. The complete linear system of hypersurfaces of degree m in L' induces a linear system Lm on V, and from the definition of Hilbert characteristic function we get X(V, m) = dim Lm + 1 from a certain m onward. Now let V't be a variety in L' of dimension r such that V' is a specialization of a positive r-cyele yr il Ln over a field k. Since addition of cycles is compatible with specialization,4 V is also a variety and the following lemma holds: LEMMA 1. A specialization of the singular locus of V over the specialization V V' with reference to k is contained in the singular locus of V'. Proof: Let W8 be a component of the singular locus of V, and let W' be a specialization of W over the specialization V -* .V' with reference to k. We have only to show that W' is contained in the singular locus of V'. Take a field of definition K of V, .V' containing k over which W8 and W' are rational. Take a generic linear Downloaded by guest on September 26, 2021 VOL. 41, 1955 MATHEMATICS: J. IGUSA 35 variety L8- over K with defining coefficients (u). Then the 0-cycle W'.L is the unique specialization of the 0-cycle WEL over the specialization (V, W) -- (V', W') with reference to k(u). Now, if W' is not contained in the singular locus of V', there exists at least one simple point P' of V' in W'OL. Let P be a point in W-L which has specialization P' in the above specialization. Now take two independent generic linear varieties Lr+l and L'-f-2 with defining coefficients (v) over the field K(u), and project V, P, V', and P' from Lf-l-2 into Lr+l obtaining VF,P. V', and P respectively. We know that (V', P') is a specialization of (1, P) over k(v). Moreover, P is multiple on 1, while P' is simple on 1'. Since 1 and 1' are hypersurfaces in LT+l, we can readily get a contradiction. In particular, if V' is nonsingular, V is also nonsingular. The following lemma is due to Chow and Nakai, but we shall give a simpler proof. LEMMA 2. If a nonsingular curve C' is a specialization of a curve C over a field k, they have the same genus. Proof: Let A and A' be the diagonals of the products C X C and C' X C'. Then (C' X C', A') is the unique specialization of (C X C, A) over the specialization C -- C' with reference to k (here we consider C X C as a subvariety of a projective space LN with N = (n + 1)2 - 1 in the usual way). Now let K be a field of definition of C, C' containing k, and let LN-3 be a generic linear variety over K in LN with defining coefficients (u). The projecting cones H and H' of A and A' with center LN-3 are hypersurfaces of the same degree, say m, in LN such that H' is the unique specialization of H over the specialization A -5 A' with reference to k(u). Now let Ho be a generic hypersurface over K(u) of degree m with defining coefficients (v). We then see that (C' X C', A', (C' X C')H', (C' X C')-Ho) is the unique specialization of (C X C, A, (C X C) H, (C X C)-Ho) over the specialization C -- C' with reference to k(u, v). If we put X = (C X C)-(Ho -H) + A and X' = (C' X C')-(Ho- H') + 'A', the intersection products A X and A'-X' are both defined. Therefore, A'-X' is the unique specialization of A X over the specialization C -> C' with reference to k(u, v). On the other hand, X is linearly equivalent to A on C X C, and X' is linearly equivalent to A' on C' X C'. There- fore,5 we have 2X(C) = deg (A-X) = deg (A'.X') = 2X(C'). The following lemma can be proved in the usual way,6 using a weak form of Kronecker-Castelnuovo's theorem, due to N6ron-Samuel.7 LEMMA 3. Let Vr be a variety in L' without multiple subvariety of dimension r - 1 (r > 2). Let D be a V-divisor which is rational over afield of definition K of V. Also, let Cm be an intersection product of V with a generic hypersurface of degree m in Ln over K. Then, if DCm is linearly equivalent to zero on Cm for every m, D itself is linearly equivalent to zero on V. Now we can prove the following: THEOREM 1. We assume that a variety V"r without multiple subvariety of dimension r - 1 is a specialization of a variety Vr over a field k. Let D be a V-divisor which is linearly equivalent to zero on V, and let D' be a specialization of D over the above specialization. Then D' is linearly equivalent to zero on V'. Proof: It follows from Lemma 1 that V has no multiple subvariety of dimension r - 1. We shall show that the assertion for r follows from the corresponding assertion for r - 1, when r > 2. In fact, take a common field of definition K of V and V' containing k over which both D and D' are rational. Let H be a generic Downloaded by guest on September 26, 2021 36 MATHEMATICS: J. IGUSA PROC. N'. A. S.

hypersurface of degree m in L' over K with defining coefficients (u). Then (V'-H, D'*H) is a specialization of (VaH, D-H) over the field k(u). We know8 that V'H is again a variety without multiple subvariety of dimension r - 2. Since D-H is linearly equivalent to zero on VAH, by induction assumption, D'-H is linearly equivalent to zero on V'aH. Since this is true for every H, by Lemma 3, D' is linearly equivalent to zero on V'. We may therefore assume that V and V' are both nonsingular curves C and C'. Moreover, it is quite easy to see that we have only to prove the following assertion. If Z is a member of Lm and if Z' is a specialization of Z over the specialization C C' with reference to k, then Z' is a member of the trace Lm' on C' of the complete linear system of hypersurfaces of degree m in L' for all m sufficiently large. Now let 4) be the everywhere regular rational mapping of L' into a projective space LN which is associated with the complete linear system of hypersurfaces of degree m in L'. Since 4) is everywhere regular, if soand p' are functions induced by 4) on C and C', then their graphs are related to the graph of 4) by rl, = rP(c x LN) and r,,f9 = F4-(C' X Lu). On the other hand, since C and C' are nonsingular, both are normal, and therefore Lm and Lm' are complete9 from a certain m onward and have the same dimension by Lemma 2. Now let L8 be the smallest linear subvariety of LN containing the projection of r1, on LN. We can find a hyperplane L'-1 of L8 such that Z = prc LrL (C X Ls')]. Let (r', L'8, L'8-1) be a specialization of (r,9, L8, L8s) over the specialization (C, Z) -- (C', Z') with reference to k. Then, first of all, we have rF = rF, by specialization theory. Moreover, since s is equal to the dimension of Lm and hence also of Lm', we see that L'8 is the smallest linear subariety of LN containing the projection of r,9' on LN. Therefore, the intersection product r,,9P(C' X L's-') is defined on C' X L'8. Hence, by specialization theory,10 we conclude: Z' = prcy [FP, (C' X L'"')], completing the proof. The following is our main theorem: THEOREM 2. If a normal variety V'r is a specialization of a variety vr, then V is also normal and they have the same Hilbert characteristic function, hence a fortiori the same arithmetic genus. Proof: We can use the same notations as in the above proof, replacing C and C' by V and V'. Since L'8 contains the projection of r1,9 on LN we get dim L. > dim Lm'. On the other hand, consider the Chow varieties U and U' of the complete linear systems IL.i and L' for m sufficiently large. Let U* be a specialization of U over the specialization V -_ V! with reference to k. Then we conclude from Theorem 1 that U* is carried by U', whence dim U = dim U* < dim U'. However, we have dim U = drn IL. >. dim L, > dimL, = dim U'; hence they are all equal. Therefore, Lm is complete whenever L' is complete. In view of the Muhley-Zariski criterion" this proves the normality of V and also that both V and V' have the same Hilbert characteristic function. COROLLARY. If two members of a maximal algebraic family of positive recycles in L' are both normal varieties, they have the same arithmetic genus. Downloaded by guest on September 26, 2021 'VOL. 41, 1955 MATHEMATICS: NICOL AND VANDIVER 37

We note that we cannot expect a better assertion thani the above one. Namely, the bunch of positive r-cycles of the same degree in L' is connected and yet two nonsingular members of this bunch may wvell have different arithmetic genera. * This work was supported by a research project at Harvard University, sponsored by the Office of Ordnance Research, United States Army, under Contract DA-19020-ORD-3100. 1 A. Weil, Variet0s Ab~liennes et courbes alg~briques ("Actualit6s Sci. et Ind.," No. 1064 [Paris: Hermann et Cie, 1948]), 61-62, Lemma 10. 2 W. L. Chow, "On the Genus of Curves of an Algebraic System," Trans. Am. Math. Soc., 65, 137-140, 1949; Y. Nakai, "On the Genus of Algebraic Curves," Kyoto Math. Mem., 27, 163-165 1952. 3To appear shortly. 4 Cf., e.g., T. Matsusaka, "Specialization of Cycles on a Projective Model," Kyoto Math. Mem., 26, 167-173 1951. 5 A. Weil, Courbes altgbriques ("Actualit6s Sci. et Ind.," No. 1041 [Paris: Hermann et Cie, 19481), 13. 6 0. Zariski, Algebraic Surfaces (Chelsea, New York, 1948), 88-89. 7A. N6ron and P. Samuel, "La Vari6th de Picard d'une variUt6 normale," Ann. de l'Institut Fourier, 4, 1-30, 1953. 8 This follows from the so-called first and second Bertini's theorems. 0. Zariski, "Complete Linear Systems on Normal Varieties and a Generalization of a Lemma of Enriques-Severi," Ann. Math., 55, 552-592, 1952. '0 Strictly speaking, the compatibility of the intersection product and the specialization is proved only in the case when the ambient variety is fixed. However, we can reduce our general case to this case by using projecting cones-. 11 Zariski, Ann. Math., 55, 563, 1952.

ON GENERATING FUNCTIONS FOR RESTRICTED PARTITIONS OF RATIONAL INTEGERS BY C. A. NICOL AND H. S. VANDIVER UNIVERSITY OF TEXAS, AUSTIN, TEXAS Communicated November 12, 1954 Introduction.-Suppose that a, . . . , a,, are nonnegative integers not all of which - are zero, and consider the function n F(z, x, n, a,, . .. , an) = Ia(1 - ,zx)a; (1) i= 1 For z = 1 or z = -1 and a, = 1, i = 1, . .. , n, we have a function extensively studied.' We propose to give in the first part of this paper a recursion relation (4) for the coefficients of the polynomial in x resulting from the expansion of the product dis- played in (1). We follow this by specializing (1) by choosing z = 1 and present a theorem concerning the symmetry of the coefficients of the polynomial obtained by expanding (1). Then, taking a, = 1 for i = 1, . .. , n, we determine some arithmetic properties of the coefficients of this polynomial. Pervading most of these relations is the von Sterneck number,2

~~(k, n) = 4? (k, n) =A~n~(kn(n)))n) Downloaded by guest on September 26, 2021