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

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 : setRandomSeed 0
 -- setting random seed to 0

o1 = 0
i2 : kk=ZZ/101

o2 = kk

o2 : QuotientRing
i3 : g=1

o3 = 1
i4 : (S, qq, R,  u, M1, M2, Mu1, Mu2)=randomNicePencil(kk,g);
i5 : gens S

o5 = {x , y , z , z , s, t}
       0   0   1   2

o5 : List
i6 : q1 = diff(S_(2*g+2),qq)

             2
o6 = x y  - z
      0 0    1

o6 : S

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

i7 : ideal u -- an isotropic space for q1 and q2

o7 = ideal (x , z , z )
             0   1   2

o7 : Ideal of S
i8 : betti Mu1, betti Mu2

             0 1         0 1
o8 = (total: 4 4, total: 4 4)
          1: 3 1      2: 1 .
          2: 1 3      3: 3 3
                      4: . 1

o8 : Sequence
i9 : Mu1*Mu2- qq*id_(target Mu1) == 0

o9 = true

See also

Ways to use randomNicePencil:

  • randomNicePencil(Ring,ZZ)

For the programmer

The object randomNicePencil is a method function.


The source of this document is in PencilsOfQuadrics.m2:1970:0.