randomNicePencil -- sets up a random example to construct Clifford algebra and representation

Synopsis

Usage:

(S, qq, R, u, M1, M2, Mu1, Mu2)=randomNicePencil(kk,g)
Inputs:
- kk, a ring, the ground field, not char 2, please!
- g, an integer, genus of the associated hyperelliptic curve.
Outputs:
- S, a ring, polynomial ring in g x's, g y's z_1,z_2 and s, t
- qq, a ring element, Element of S, quadratic in x_0..z_2 and linear in s,t
- R, a ring, polynomial ring kk[s,t]
- u, a matrix, 1 x g+2 row matrix with entries x_0..x_{(g-1)}, z_1,z_2; generators of the ideal of an isotropic subspace for qq
- M1, a matrix,
- M2, a matrix, Matrices over S such that M1*M2 = qq times an identity of size 2^{2g+1}
- Mu1, a matrix,
- Mu2, a matrix, Matrices over S such that Mu1*Mu2 = qq times an identity of size 2^{g+1}

Description

Chooses a random example of a pencil of quadrics qq = s*q1+t*q2 with a fixed isotropic subspace (defined by ideal u) and a fixed corank one quadric in normal form q1.

When called with no arguments it prints a usage message.

The variables of S that are entries of X:= matrix \{\{x_0..y_{(g-1)},z_1,z_2\}\} \, represent coordinates on PP_R^{2g+1}.

M1, M2 are consecutive high syzygy matrices in the minimal (periodic) resolution of kk[s,t] = S/(ideal X) as a module over S/qq. These are used to construct the Clifford algebra of qq.

Mu1, Mu2 are consecutive high syzygy matrices in the minimal (periodic) resolution of S/(ideal u) as a module over S/qq. These are used to construct a Morita bundle between the even Clifford algebra of qq and the hyperelliptic curve branched over the degeneracy locus of the pencil,

\{(s,t) | s*q1+t*q2 is singular\} \subset PP^1.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing

i2 : g=1

o2 = 1

i3 : (S, qq, R,  u, M1, M2, Mu1, Mu2)=randomNicePencil(kk,g);

i4 : gens S

o4 = {x , y , z , z , s, t}
       0   0   1   2

o4 : List

i5 : q1 = diff(S_(2*g+2),qq)

             2
o5 = x y  - z
      0 0    1

o5 : S

a quadratic form of corank 1 (corresponding to a branch point of E-->PP^1 in normal form.

i6 : ideal u -- an isotropic space for q1 and q2

o6 = ideal (x , z , z )
             0   1   2

o6 : Ideal of S

i7 : betti Mu1, betti Mu2

             0 1         0 1
o7 = (total: 4 4, total: 4 4)
          1: 3 1      2: 1 .
          2: 1 3      3: 3 3
                      4: . 1

o7 : Sequence

i8 : Mu1*Mu2- qq*id_(target Mu1) == 0

o8 = true

Ways to use randomNicePencil :

randomNicePencil(Ring,ZZ)

For the programmer

The object randomNicePencil is a method function.

randomNicePencil -- sets up a random example to construct Clifford algebra and representation

Synopsis

Description

See also

Ways to use randomNicePencil :

For the programmer