How to Calculate a -hilb C
How to Calculate a -hilb C
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如何计算 -hilb C
DOI：
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发表时间：
2001
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通讯作者：
A. Reid
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文献类型：
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作者：
A. Reid
中村[N]对有限子群G ∈ SL（3，C）引入了G-Hilbert格式G-HilbC 3，并证明了它是商C3/G的一个可分解.他证明了这一点的对角线阿贝尔群A通过引入一个明确的算法，计算A-Hilb C3。这篇笔记计算A-Hilb C3要简单得多，就连分数加上等边三角形的规则镶嵌的乐趣而言。1结果说明1.1初级单形与三个牛顿多边形设A ∈ SL（3，C）是作用在C上的对角子群。将A的所有元素以1 r（a1，a2，a3）的形式生成的超晶格写成L <$Z。初级单形有3个顶点e1 =（1，0，0），e2 =（0，1，0）和e3 =（0，0，1）。记R为所张成的仿射平面，Z 2 = L <$R 2为相应的仿射格。依次以每个ei为原点，构造牛顿多边形，得到的牛顿多边形为每个ei中格点的凸船体（见图1.a）：fi，0，fi，1，fi，2，. . .其中fi，0是沿[ei，ei-1]的边沿着的基元向量，fi，ki+1是沿[ei，ei+1]的边沿着的基元向量。(The指数i，i ± 1是循环的。此外，由于ei是原点，符号fi，j表示Ei的格点和对应的
Nakamura [N] introduced the G-Hilbert scheme G-Hilb C3 for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-Hilb C3. This note calculates A-Hilb C3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilateral triangles. 1 Statement of the result 1.1 The junior simplex and three Newton polygons Let A ⊂ SL(3, C) be a diagonal subgroup acting on C. Write L ⊃ Z for the overlattice generated by all the elements of A written in the form 1 r (a1, a2, a3). The junior simplex ∆ (compare [IR], [R]) has 3 vertexes e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 = (0, 0, 1). Write R∆ for the affine plane spanned by ∆, and Z 2 ∆ = L ∩ R 2 ∆ for the corresponding affine lattice. Taking each ei in turn as origin, construct the Newton polygons obtained as the convex hull of the lattice points in ∆ ei (see Figure 1.a): fi,0, fi,1, fi,2, . . . , fi,ki+1, (1.1) where fi,0 is the primitive vector along the side [ei, ei−1], and fi,ki+1 that along [ei, ei+1]. (The indices i, i ± 1 are cyclic. Also, since ei is the origin, the notation fi,j denotes both the lattice point of ∆ and the corresponding