Zero-Dimensional Branches of Rank One on Algebraic Varieties

MacLane, Saunders; Schilling, O. F. G.

doi:10.2307/1968935

Cited by 46 publications

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“…Let us consider the nonhomogeneous coordinates ^i^Vi/vo-Should the center Wbe at finite distance with respect to these coordinates, we must have 770^0 on W. But then, by the remark just made, v(vi/vo) =■(), i = 0, 1, • ■ ■ , n, and the entire coordinate ring 0 must be contained in the valuation ring R" of v. where/and g (4) Chain theorem rings with the property that their non-units form an ideal have been called by Krull "Slellenringen" (see Krull [3]). We propose the translation:…”

Section: Concretelymentioning

confidence: 99%

“…Hence f is a non-unit in Rv, and, in view of our assumption that g is not zero on W, this is only possible if /=0 on W, q.e.d. 4. Existence theorems for valuations with a preassigned center.…”

Section: Concretelymentioning

confidence: 99%

“…The valuation v of 2 induced by v' will have center Wo, dimension 5 and rank 1. (13) The existence of v' is proved in the joint paper by MacLane and Schilling [4]. (14) This is implied by the fundamental theorem on principal orders which states that an integrally closed integral domain (not a field) is the intersection of the valuation rings which contain it (see Krull [2,p.…”

Section: Theoremmentioning

confidence: 99%

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Foundations of a General Theory of Birational Correspondences

Zariski¹

1943

Transactions of the American Mathematical Society

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In our papers dealing with the reduction of singularities of an algebraic surface (see [8, ll]), we were forced to devote a good deal of space to certain properties of birational correspondences for which we could find no general proofs in the literature. These properties were of a general character and therefore could not be regarded as part of the reduction proof proper, although they did play an auxiliary role in the proof. A similar situation arose in our reduction proof for three-dimensional varieties (not yet published), but in this case the amount of preliminary general material on birational correspondences used in the proof was even larger and was out of proportion to the length of the reduction proof proper. It thus became increasingly clear that the procedure of treating general questions of birational correspondences only as and when these questions come up in connection with various steps of the reduction process, could no longer be followed in the case of higher varieties. Instead it seemed necessary-and also worthwhile for its own saketo develop systematically in a separate paper the fundamental concepts and theorems of the theory of birational correspondences, and to do this in as general a fashion as possible. This we propose to do in the present paper. We deal here with algebraic varieties, with or without singularities, over an arbitrary ground field (of characteristic zero or p). It is difficult to say which of our results are entirely novel and which are not. Since many of the results hold only for normal varieties, they would appear to be novel inasmuch as our concept of a normal variety is new. On the other hand, most of our results were known for nonsingular models. It is perhaps correct to say that the novelty of the present investigation consists in showing that most of the known properties of birational correspondences between nonsingular varieties remain true more generally for normal varieties.

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Section: Concretelymentioning

confidence: 99%

Section: Concretelymentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Foundations of a General Theory of Birational Correspondences

Zariski¹

1943

Transactions of the American Mathematical Society

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“…In [6], Mac Lane and Schilling construct the value group corresponding to such a valuation; that is, they describe the image of the nonzero elements of the function field k(x, y) under a valuation that is given by a series expansion. The purpose of this paper is to illuminate the behavior of the image Λ = {v(f ) : f ∈ k[x, y] } (called the value monoid) of the nonzero elements of the underlying polynomial ring k [x, y] under such a valuation.…”

Section: Introductionmentioning

confidence: 99%

The growth of valuations on rational function fields in two variables

Mosteig

Sweedler

2004

Proc. Amer. Math. Soc.

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Abstract. Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k [x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let Λn denote the images under the valuation v of all nonzero polynomials f ∈ k [x, y] of at most degree n in the variable y. We construct a bound for the growth of Λn with respect to n for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

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“…It is well known (2) Moreover v0 can be taken such that fl'(x,)säO, *=1, • • • , k. In fact, since v(xi)^0 it follows that the f-residues of the are finite, and hence there exists a v0 such that »o(£,)=0, whence f'(x,) = 0. £5 is then contained in S3', and this completes the proof of our assertion.…”

mentioning

confidence: 99%

Valuation ideals in polynomial rings

Seidenberg¹

1945

Trans. Amer. Math. Soc.

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Zero-Dimensional Branches of Rank One on Algebraic Varieties

Cited by 46 publications

References 0 publications

Foundations of a General Theory of Birational Correspondences

Foundations of a General Theory of Birational Correspondences

The growth of valuations on rational function fields in two variables

Valuation ideals in polynomial rings

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