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- W1519772113 abstract "where Xs denotes the smooth locus of X, consider: Problem 1. Find all affine rulings of X. Problem 2. Find all pairs of curves C1, C2 on X such that X n (C1 [ C2) is isomorphic to P2 minus two lines. Problem 3. Find all curves C in X such that (Xs n C) = 1. This paper investigates Problem 1 for an arbitrary X satisfying (†). We define (Definition 1.14) the notion of a “basic” affine ruling of X and our main results describe how to construct all affine rulings of X, assuming that the basic ones are known. In the case where X is a weighted projective plane, the basic affine rulings of X are given in [6]; the present paper and [6] therefore constitute a solution to Problem 1 in that case. Problem 3 (with X = P2) has been considered by several authors ([8], [9], [18], [19], [14]). In his review of [14] (see MR 82k:14013), M. H. Gizatullin mentions some unpublished examples found by V. I. Danilov and himself, and which seem to correspond to the list of basic affine rulings of P2. The case X = P2 was finally solved in [10]. Our generalization to weighted projective planes seems to be new, as well as our method—valid for any X satisfying (†)—which reduces the general problem to the determination of the basic affine rulings. Let us briefly indicate how problems 1–3 are related to each other. Consider the stronger condition (‡) on a surface X:" @default.
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- W1519772113 date "2001-03-01" @default.
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- W1519772113 title "AFFINE RULINGS OF NORMAL RATIONAL SURFACES" @default.
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- W1519772113 doi "https://doi.org/10.18910/10530" @default.
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