An arithmetic theory of adjoint plane curves
An arithmetic theory of adjoint plane curves
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伴随平面曲线的算术理论
DOI：
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发表时间：
1952
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通讯作者：
D. Gorenstein
中科院分区：
文献类型：
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作者：
D. Gorenstein
介绍。在经典代数几何中，不可约平面曲线的伴随曲线是研究曲线几何的重要工具。本文给出伴随曲线理论的代数-算术发展，并将经典结果推广到具有任意奇异点的不可约平面曲线上。我们对曲线上给定奇点处伴随条件的定义是用点的局部环与其积分闭合之间的导体来表示的。伴随曲线的基本性质，然后由相应的导体性质导出。伴随曲线的一个最深刻和最重要的性质是，在m阶曲线上，m -3阶的伴随曲线切断了完整的正则级数。这个性质等价于这样一个事实，即伴随级数的固定分量的程度是伴随曲线对足够高阶(1)曲线施加的条件数目的两倍。对于这个命题，我们将给出两个不同而独立的证明。第一个证明是直接的，基于对给定曲线奇点的详细分析。第一部分所讨论的这种分析同样适用于代数数域，我们的处理将包括这种情况和单变量代数函数域的情况。第二种证明更加间接，依赖于黎曼-洛克定理和对第一类微分的经典表示定理的推广。这个将在第二部分给出的证明，只适用于函数场是在地场上可分离生成的平面曲线。
Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor. The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m -3 cut out the complete canonical series. This property is equivalent to the fact that the degree of the fixed component of the adjoint series is twice the number of conditions which the adjoint curves impose on the curves of sufficiently high order(1). We shall give two distinct and independent proofs of this proposition. The first proof is a direct one, based upon a detailed analysis of the singularities of the given curve. This analysis, to which part I is devoted, applies equally well to algebraic number fields, and our treatment will include this case with that of algebraic function fields of one variable. The second proof is more indirect, depending upon the Riemann-Roch theorem and a generalization of the classical representation theorem of the differentials of the first kind. This proof, which will be given in part II, holds only for plane curves whose function field is separably generated over the ground field.