Top
Back: C.4 Characteristic sets
Forward: C.6 Toric ideals and integer programming
FastBack: Appendix C Mathematical background
FastForward: Appendix D SINGULAR libraries
Up: Appendix C Mathematical background
Top: 1 Preface
Contents: Table of Contents
Index: F Index
About: About This Document

C.5 Gauss-Manin connection

Let f:(Cn+1,0) (C,0) be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by f. Let O = C[x0,,xn](x0,,xn) be the local ring at the origin and Jf the Jacobian ideal of f.

A Milnor representative of f defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of μ n-spheres. The n-th cohomology bundle is a flat vector bundle of dimension n and carries a natural flat connection with covariant derivative t. The monodromy operator is the action of a positively oriented generator of the fundamental group of the puctured disc on the Milnor fibre. Sections in the cohomology bundle of moderate growth at 0 form a regular D = C{t}[t]-module G, the Gauss-Manin connection.

By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of G. This factors through an inclusion of the Brieskorn lattice H“ := ΩCn+1,0n+1∕df dΩCn+1,0n1 in G.

The D-module structure defines the V-filtration V on G by V α := βαC{t}ker(t∂t β)n+1. The Brieskorn lattice defines the Hodge filtration F on G by Fk = tkH“ which comes from the mixed Hodge structure on the Milnor fibre. Note that F1 = H‘.

The induced V-filtration on the Brieskorn lattice determines the singularity spectrum Sp by Sp(α) := dimCGrV αGr0FG. The spectrum consists of μ rational numbers α1,μ such that e2πiα1,,e2πiαμ are the eigenvalues of the monodromy. These spectral numbers lie in the open interval (1,n), symmetric about the midpoint (n 1)2.

The spectrum is constant under μ-constant deformations and has the following semicontinuity property: The number of spectral numbers in an interval (a,a + 1] of all singularities of a small deformation of f is greater or equal to that of f in this interval. For semiquasihomogeneous singularities, this also holds for intervals of the form (a,a + 1).

Two given isolated singularities f and g determine two spectra and from these spectra we get an integer. This integer is the maximal positive integer k such that the semicontinuity holds for the spectrum of f and k times the spectrum of g. These numbers give bounds for the maximal number of isolated singularities of a specific type on a hypersurface X Pn of degree d: such a hypersurface has a smooth hyperplane section, and the complement is a small deformation of a cone over this hyperplane section. The cone itself being a μ-constant deformation of x0d + + xnd = 0, the singularities are bounded by the spectrum of x0d + + xnd.

Using the library gaussman.lib one can compute the monodromy, the V-filtration on H∕H‘, and the spectrum.

Let us consider as an example f = x5 + x2y2 + y5 . First, we compute a matrix M such that exp(2πiM) is a monodromy matrix of f and the Jordan normal form of M :

  LIB "gaussman.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  list l=monodromy(f);
  matrix M=jordanmatrix(l[1],l[2],l[3]);
  print(M);
→ 1/2,0,  0,   0,   0,   0,   0,0,    0,    0,    0,   
→ 1,  1/2,0,   0,   0,   0,   0,0,    0,    0,    0,   
→ 0,  0,  7/10,0,   0,   0,   0,0,    0,    0,    0,   
→ 0,  0,  0,   7/10,0,   0,   0,0,    0,    0,    0,   
→ 0,  0,  0,   0,   9/10,0,   0,0,    0,    0,    0,   
→ 0,  0,  0,   0,   0,   9/10,0,0,    0,    0,    0,   
→ 0,  0,  0,   0,   0,   0,   1,0,    0,    0,    0,   
→ 0,  0,  0,   0,   0,   0,   0,11/10,0,    0,    0,   
→ 0,  0,  0,   0,   0,   0,   0,0,    11/10,0,    0,   
→ 0,  0,  0,   0,   0,   0,   0,0,    0,    13/10,0,   
→ 0,  0,  0,   0,   0,   0,   0,0,    0,    0,    13/10

Now, we compute the V-filtration on H∕H‘ and the spectrum:

  LIB "gaussman.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  list l=vfilt(f);
  print(l[1]);
→ -1/2,
→ -3/10,
→ -1/10,
→ 0,
→ 1/10,
→ 3/10,
→ 1/2
  print(l[2]);
→ 1,2,2,1,2,2,1
  print(l[3]);
→ [1]:
→    _[1]=gen(11)
→ [2]:
→    _[1]=gen(10)
→    _[2]=gen(6)
→ [3]:
→    _[1]=gen(9)
→    _[2]=gen(4)
→ [4]:
→    _[1]=gen(5)
→ [5]:
→    _[1]=gen(3)
→    _[2]=gen(8)
→ [6]:
→    _[1]=gen(2)
→    _[2]=gen(7)
→ [7]:
→    _[1]=gen(1)
  print(l[4]);
→ y5,
→ y4,
→ y3,
→ y2,
→ xy,
→ y,
→ x4,
→ x3,
→ x2,
→ x,
→ 1

Here l[1] contains the spectral numbers, l[2] the corresponding multiplicities, l[3] a C -basis of the V-filtration on H∕H‘ in terms of the monomial basis of O∕Jf∼=H∕H‘ in l[4].

If the principal part of f is C-nondegenerate, one can compute the spectrum using the library spectrum.lib. In this case, the V-filtration on H“ coincides with the Newton-filtration on H“ which allows to compute the spectrum more efficiently.

Let us calculate one specific example, the maximal number of triple points of type 6 on a surface X P3 of degree seven. This calculation can be done over the rationals. So choose a local ordering on Q[x,y,z] . Here we take the negative degree lexicographical ordering which is denoted ds in SINGULAR:

ring r=0,(x,y,z),ds;
LIB "spectrum.lib";
poly f=x^7+y^7+z^7;
list s1=spectrumnd( f );
s1;
→ [1]:
→    _[1]=-4/7
→    _[2]=-3/7
→    _[3]=-2/7
→    _[4]=-1/7
→    _[5]=0
→    _[6]=1/7
→    _[7]=2/7
→    _[8]=3/7
→    _[9]=4/7
→    _[10]=5/7
→    _[11]=6/7
→    _[12]=1
→    _[13]=8/7
→    _[14]=9/7
→    _[15]=10/7
→    _[16]=11/7
→ [2]:
→    1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1

The command spectrumnd(f) computes the spectrum of f and returns a list with six entries: The Milnor number μ(f), the geometric genus pg(f) and the number of different spectrum numbers. The other three entries are of type intvec. They contain the numerators, denominators and multiplicities of the spectrum numbers. So x7 + y7 + z7 = 0 has Milnor number 216 and geometrical genus 35. Its spectrum consists of the 16 different rationals
3_ 7, 4 7, 5 7, 6 7, 1 1, 8 7, 9 7, 10 7 , 11 7 , 12 7 , 13 7 , 2 1, 15 7 , 16 7 , 17 7 , 18 7
appearing with multiplicities
1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.

The singularities of type 6 form a μ-constant one parameter family given by x3 + y3 + z3 + λxyz = 0, λ327.

Therefore they have all the same spectrum, which we compute for x3 + y3 + z3.

poly g=x^3+y^3+z^3;
list s2=spectrumnd(g);
s2;
→ [1]:
→    8
→ [2]:
→    1
→ [3]:
→    4
→ [4]:
→    1,4,5,2
→ [5]:
→    1,3,3,1
→ [6]:
→    1,3,3,1

Evaluating semicontinuity is very easy:

semicont(s1,s2);
→ 18

This tells us that there are at most 18 singularities of type 6 on a septic in P3. But x7 + y7 + z7 is semiquasihomogeneous (sqh), so we can also apply the stronger form of semicontinuity:

semicontsqh(s1,s2);
→ 17

So in fact a septic has at most 17 triple points of type 6.

Note that spectrumnd(f) works only if f has nondegenerate principal part. In fact spectrumnd will detect a degenerate principal part in many cases and print out an error message. However if it is known in advance that f has nondegenerate principal part, then the spectrum may be computed much faster using spectrumnd(f,1).


Top Back: C.4 Characteristic sets Forward: C.6 Toric ideals and integer programming FastBack: Appendix C Mathematical background FastForward: Appendix D SINGULAR libraries Up: Appendix C Mathematical background Top: 1 Preface Contents: Table of Contents Index: F Index About: About This Document
            User manual for Singular version 2-0-4, October 2002, generated by texinfo.