完备相交空间曲线的判别式
Discriminants of complete intersection space curves
摘要：
本文提出了一种判别三维射影空间中完全交线的新方法。借助于结式理论，我们首先证明了一个新的公式，它使我们能够在任何交换环上，特别是在任何特征环上毫无歧义地定义这个判别式。该公式还提供了一种新的方法来有效地评估和计算这个判别式，而不需要像著名的凯莱技巧那样引入新的变量。在此基础上，我们得到了新的性质和计算规则，如协方差和不变性公式。最后，我们证明了我们的判别式定义满足预期的几何性质，从而产生了一个有效的光滑性准则的完整的相交空间曲线。实际上，我们表明，在一般的设置，它是定义方程的判别计划，如果基环被假定为是一个唯一的因式分解域。
英文摘要：
In this paper, we develop a new approach to the discrimi-nant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discrimi-nant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discrimi-nant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
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