ON THE BIRATIONAL EQUIVALENCE OF CURVES UNDER SPECIALIZATION

Let k be a field with a discrete valuation, let p be the maximal prime ideal in the valuation ring on k, and let be the residue field of k. Let Sⁿ be the projective space of dimension n in the algebraic geometry over a universal domain containing k, and denote by the projective space of dimension n over a universal domain containing ; if Z is any cycle in Sⁿ, rational over k, we shall denote by the cycle obtained from Z by reduction modulo p, in the sense of Shimura [4], and we shall call the specialization of Z (with respect to the given valuation of k). This definition applies also in case Z is a variety, which is then to be considered as a prime cycle; the the specialization is then a positive cycle, but in general not necessarily a variety. According to the generalization of the Principle of Degeneration recently proved by Chow, the specialization of a variety is always connected, but we shall not need this fact here…