An Algorithm to Compute the Equations of Tangent Cones
An Algorithm to Compute the Equations of Tangent Cones
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一种计算相切圆锥方程的算法
DOI：
10.1007/3-540-11607-9_18
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发表时间：
1982
期刊：
影响因子：
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通讯作者：
Ferdinando Mora
中科院分区：
文献类型：
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作者：
Ferdinando Mora
代数簇在奇点(以wlog为原点)上的切锥是该点附近代数簇的最佳逼近锥。从代数的观点来看，多项式环有一个自然的等级，在这个等级下，多项式分裂成齐次形式：多项式的初始形式是出现在这种分裂中的最小次数的齐次形式。簇V的切锥则是由理想定义V中所有多项式的初始形式的理想所定义的簇。最近，人们对切锥产生了一些兴趣，研究切锥是为了对奇点进行分类；特别是，局部代数性质(如Cohen-Macaulay，Serre，Gorenstein)从簇到它的切锥或相反的传递已经被彻底地研究(见文献[3])。解决这些问题的方法显然需要理论考虑，而不是蛮力计算；然而，切锥计算经常是需要的，至少在构造例子时是这样。
The tangent cone of an algebraic variety in a singular point (wlog the origin) is the cone which is the best approximation of the variety near the point. From an algebraic point of view, polynomial rings have a natural grading, under which polynomials split into homogeneous forms: the initial form of a polynomial is the homogeneous form of least degree which appears in this splitting. The tangent cone of a variety V is then the variety defined by the ideal of all the initial forms of polynomials in the ideal defining V.Recently some interest arose in tangent cones, which are studied in order to classify singularities; in particular, the passage of local algebraic properties (such as Cohen-Macaulay, Serre, Gorenstein) from a variety to its tangent cone or viceversa have been thoroughly investigated (see the references in [3]). The approach to these questions obviously requires theoretical considerations and not brute force computation; nevertheless tangent cone computations are often needed, at least in the construction of examples.