# 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, , Element of S, quadratic in x_0..z_2 and linear in s,t
• R, a ring, polynomial ring kk[s,t]
• u, , 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, ,
• M2, , Matrices over S such that M1*M2 = qq times an identity of size 2^{2g+1}
• Mu1, ,
• Mu2, , 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