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C.5 Gauss-Manin connection

Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by $f$. Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring at the origin and $J_f$ 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 $\mu$ $n$-spheres. The $n$-th cohomology bundle is a flat vector bundle of dimension $n$ and carries a natural flat connection with covariant derivative $\partial_t$. The monodromy operator is the action of a positively oriented generator of the fundamental group of the punctured disc on the Milnor fibre. Sections in the cohomology bundle of moderate growth at $0$ form a regular $D=C\{t\}[\partial_t]$-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 $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in $G$.

The $D$-module structure defines the V-filtration $V$ on $G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$. The Brieskorn lattice defines the Hodge filtration $F$ on $G$ by $F_k=\partial_t^kH''$ which comes from the mixed Hodge structure on the Milnor fibre. Note that $F_{-1}=H'$.

The induced V-filtration on the Brieskorn lattice determines the singularity spectrum $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$. The spectrum consists of $\mu$ rational numbers $\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi i\alpha_\mu}$ 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 $\mu$-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 than 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 $X\subset{P}^n$ 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 $\mu$-constant deformation of $x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of $x_0^d+\dots+x_n^d$.

Using the library gmssing.lib one can compute the monodromy, the V-fitration on $H''/H'$, and the spectrum.

Let us consider as an example $f=x^5+x^2y^2+y^5$. First, we compute a matrix $M$ such that $\exp(2\pi iM)$is a monodromy matrix of $f$ and the Jordan normal form of $M$:

 
  LIB "mondromy.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  matrix M=monodromyB(f);
  print(M);
==> 11/10,0,    0,    0,    0,   0,-1/4,0,   0,   0,  0,   
==> 0,    13/10,0,    0,    0,   0,0,   15/8,0,   0,  0,   
==> 0,    0,    13/10,0,    0,   0,0,   0,   15/8,0,  0,   
==> 0,    0,    0,    11/10,-1/4,0,0,   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,  3/5, 
==> 0,    0,    0,    0,    0,   0,9/10,0,   0,   0,  0,   
==> 0,    0,    0,    0,    0,   0,0,   7/10,0,   0,  0,   
==> 0,    0,    0,    0,    0,   0,0,   0,   7/10,0,  0,   
==> 0,    0,    0,    0,    0,   0,0,   0,   0,   1,  -2/5,
==> 0,    0,    0,    0,    0,   0,0,   0,   0,   5/8,0

Now, we compute the V-fitration on $H''/H'$ and the spectrum:

 
  LIB "gmssing.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  list l=vfilt(f);
  print(l[1]); // spectral numbers
==> -1/2,
==> -3/10,
==> -1/10,
==> 0,
==> 1/10,
==> 3/10,
==> 1/2
  print(l[2]); // corresponding multiplicities
==> 1,
==> 2,
==> 2,
==> 1,
==> 2,
==> 2,
==> 1 
  print(l[3]); // vector space of i-th graded part
==> [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]); // monomial vector space basis of H''/s*H''
==> y5,
==> y4,
==> y3,
==> y2,
==> xy,
==> y,
==> x4,
==> x3,
==> x2,
==> x,
==> 1
  print(l[5]); // standard basis of Jacobian ideal
==> 2x2y+5y4,
==> 5x5-5y5,
==> 2xy2+5x4,
==> 10y6+25x3y4
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 $O/J_f\cong H''/H'$in l[4] (separated by degree).

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 $\tilde{E}_6$ on a surface $X\subset{P}^3$of degree seven. This calculation can be done over the rationals. We choose a local ordering on $Q[x,y,z]$. Here we take the negative degree lexicographical ordering, in SINGULAR denoted by ds:

 
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 $\mu(f)$, the geometric genus $p_g(f)$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 $x^7+y^7+z^7=0$has Milnor number 216 and geometrical genus 35. Its spectrum consists of the 16 different rationals
${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1}, {8 \over 7}, {... ...3 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7}, {18 \over 7}$
appearing with multiplicities
1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.

The singularities of type $\tilde{E}_6$ form a $\mu$-constant one parameter family given by $x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.Therefore they have all the same spectrum, which we compute for $x^3+y^3+z^3$.

 
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 $\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$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 $\tilde{E}_6$.

Note that spectrumnd(f) works only if $f$ has a 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).