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NumericalGodeaux :: verifyThmHypLocus

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

Synopsis

Description

The hyperelliptic locus $V_{hyp}$ in $Q \subset \mathbb{P}^{11}$ is is birational to a product of a Hirzebruch surface $F$ with 3 copies of $\mathbb{P}^{1}$. The procedure returns a ring map corresponding to the rational map $$ \phi: F \ \times \ \mathbb{P}^1 \ \times \ \mathbb{P}^1 \ \times \ \mathbb{P}^1 \ \rightarrow \ V_{hyp} \ \subset \ \mathbb{P}^{11}.$$ 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 :

For the programmer

The object verifyThmHypLocus is a method function.