verifyThmHypLocus -- print commands which verify the assertions on the hyperelliptic locus

Synopsis

Usage:
(Jhyp,paraHyp) = verifyThmHypLocus(kk)
Inputs:
- kk, a ring, the ground field, mainly QQ
Outputs:
- Jhyp, an ideal, defining the hyperelliptic locus
- paraHyp, a ring map, defininig the precomputed parametrization of the hyperelliptic loci

Description

The hyperelliptic locus V_hyp in Q ⊂ℙ¹¹ is is birational to a product of a Hirzebruch surface F with 3 copies of ℙ¹. The procedure returns a ring map corresponding to the rational map

φ: F × ℙ¹ × ℙ¹ × ℙ¹ → V_hyp ⊂ ℙ¹¹.

This function prints the commands which check that the assertion of Thm 2.9 in [F.-O. Schreyer and I. Stenger, Marked Godeaux surfaces with special bicanonical fibers. https://arxiv.org/pdf/2201.12065.pdf] are true.

i1 : (Jhyp,paraHyp) =  verifyThmHypLocus(QQ);

   betti res Jhyp
   paraHyp(Jhyp)==0
   rank jacobian matrix paraHyp == dim Jhyp
   pt=random(kk^1,kk^(#support matrix paraHyp),Height=>10)
   pt1=(vars ring Jhyp)*syz sub(matrix paraHyp,pt);
   fiber=ideal paraHyp(pt1);
   baseLocus=ideal matrix paraHyp;
   cBL=decompose baseLocus;
   scan(cBL,c-> fiber=saturate(fiber,c))
   dim fiber == 5
   fiber
   degree fiber==degree ideal gens target paraHyp
   sub(fiber,pt)

Ways to use `verifyThmHypLocus` :

verifyThmHypLocus(Ring)

verifyThmHypLocus -- print commands which verify the assertions on the hyperelliptic locus

Synopsis

Description

Ways to use verifyThmHypLocus :

Ways to use `verifyThmHypLocus` :