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PrymCanonicalCurves :: getSepticOfGenus8With2Torsion

getSepticOfGenus8With2Torsion -- Construct a plane septic curve C of geometric genus 8 together with a 2-torsion divisor

Synopsis

Description

We construct a curve C of genus 8 with a g27 together with a 2-torsion divisor. The model C’ of C in P2 has 7 double points. The divisor D0 will consist of eight points in which C’ and a rational quartic Q are tangent. We choose Q and C’ such that they have three double points in common. Since 7*4-3*22==2*8 there are no more intersection points. The divisor D1 will be given by the 8=7*2-3*2 additional intersection points of C’ with a conic defined by g1 through the 3 singular points. Note that D0-D1 is 2-torsion in Pic C.

i1 : p=10007;kk=ZZ/p;
i3 : S=kk[x_0..x_2]

o3 = S

o3 : PolynomialRing
i4 : time (I,D)=getSepticOfGenus8With2Torsion(S);
     -- used 0.386529 seconds
i5 : dim I, degree I

o5 = (2, 7)

o5 : Sequence
i6 : geometricGenus=genus I - degree ideal jacobian I

o6 = 8
Key idea of the construction is the following: Choose a parametrization of the quartic Q first, together with 6 of the desired 8 points of tangency. The linear system of septics double at at the 7=3+4 points and tangent to Q in the 6 given point is a net. We find the desired C in the net by the condition that the remaining 4 points on P1 are defined by a square. More precisely, we consider the intersection of the net with the Veronese surface of squares inside P(H0(P1,O(4)))*. This are 4 points and for random choice we expect a kk-rational intersection points in 62.5% of the cases. Thus a probabilistic approach works.

Ways to use getSepticOfGenus8With2Torsion :

  • getSepticOfGenus8With2Torsion(PolynomialRing)