Calculating discriminants by higher direct images
Calculating discriminants by higher direct images
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通过更高的直接图像计算判别式
DOI：
10.1090/s0002-9947-1994-1184118-6
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
J. Weyman
中科院分区：
文献类型：
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作者：
J. Weyman
利用同调代数，对非奇异射影簇X上的任意线丛Y，构造了行列式等于对偶簇XV的方程的复数IF(S).这推广了Gelfand，Kapranov和Zlevinski定义的Cayley-Koszul复形。给出了用复形F(Y)表示的XV的余维和次数的公式。在文章的第二部分中，将一般技巧应用于经典判别式和超行列式。本文由两部分组成。在第一部分(？？1和2)中，我们给出了Gelfand，Kapranov和Zlevinski所考虑的广义判别式的计算方法。[G-K-Z])。对任意非奇异射影簇X定义广义判别式，作为对偶簇XV的定义方程。本文介绍的计算这些方程的方法采用了正交化结构和高阶直接映象技术。对于X上的每个线丛Y，我们定义了正则复形IF(S)，它是通过下推与对偶簇XV的正则去奇化有关的扭曲的Koszul分解而获得的。复数IF(Y)允许显式计算判别式的次数，并根据与IF(Y)相关的Buchsbaum-Eisenbud乘子给出它的表达式。一般技巧在？1中描述，其中证明了所有的一般公式。第2节讨论G-鉴别式的情况。当非奇异簇X是约化群G的不可约表示中最高权向量的G-轨道时，它们就会发生。在第二部分(？？3和4)中，我们研究了判别式的行列式表达式。它们自然地出现在复合体F(Y)的上下文中。如果复数1F(2‘)只有两个项，则它退化为一个矩阵。在这种情况下，判别式成为该矩阵的行列式。第三节讨论m次齐次多项式的判别式，经典的公式是两个变量(Sylvester和Bezout公式)和三个变量的情况，在Salmon[S]一书中讨论了这一点。在？3中，我们给出了这样一个公式的列表。在两个变量的情况下，该列表包括西尔维斯特表达式(？3.2，m任意，a=-1)和Bezout表达式(？3.2，m任意，a=m 2)。此外，这些公式还表现为一系列类似类型行列式的极端情况。第3.3节讨论三个变量的情况。由编辑于1992年3月27日收到，并以修订后的形式于1992年10月5日收到。1991年数学科目分类。初级13D25、14N05；次级13D02、14M15、15A72。(？)1994美国数学学会0002-9947/94每页1美元+25美元
The author uses the homological algebra to construct for any line bundle Y on a nonsingular projective variety X the complex IF(S) whose determinant is equal to the equation of the dual variety Xv . This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of Xv in terms of complexes F(Y) are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants. This paper consists of two parts. In the first part (?? 1 and 2) we give the method for calculating generalized discriminants considered by Gelfand, Kapranov and Zelevinski (cf. [G-K-Z]). The generalized discriminant is defined for any nonsingular projective variety X as the defining equation of the dual variety XV. The method for calculating these equations introduced here uses the conormal construction and the technique of higher direct images. For each line bundle Y on X we define the canonical complex IF(S), obtained by pushing down the twisted Koszul resolutions associated to the canonical desingularisation of the dual variety XV. The complex IF(Y) allows to calculate explicitly the degree of the discriminant and give the expression for it in terms of Buchsbaum-Eisenbud multipliers associated to IF(Y). The general technique is described in ? 1 where all general formulas are proved. Section 2 deals with the case of G-discriminants. They occur when the nonsingular variety X is the G-orbit of the highest weight vector in the irreducible representation of the reductive group G. In the second part (??3 and 4) we investigate the determinantal expressions of the discriminants. They arise naturally in the context of the complexes F(Y) . If the complex 1F(2') has only two terms, it reduces to a matrix. In such cases the discriminant becomes the determinant of this matrix. Section 3 deals with the case of the discriminant of a homogeneous polynomial of degree m. Classically such formulas were known for the case of two variables (Sylvester and Bezout formulas) and three variables, which is treated in the book of Salmon [S]. In ?3 we give a list of such formulas. In the case of two variables this list includes the Sylvester expression (?3.2, m arbitrary, a = -1) and Bezout expression (?3.2, m arbitrary, a = m 2). Moreover, these formulas appear as the extreme cases of the whole series of determinantal expressions of similar type. Section 3.3 deals with the case of three variables. Received by the editors March 27, 1992 and, in revised form, October 5, 1992. 1991 Mathematics Subject Classification. Primary 13D25, 14N05; Secondary 13D02, 14M15, 15A72. (?) 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page