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gap> pg1 := PG(1, 7);
ProjectiveSpace(1, 7)
gap> pg3 := PG(3, 7);
ProjectiveSpace(3, 7)
gap> points := Points(pg1);
<points of ProjectiveSpace(1, 7)>
gap> coords := List(points, Coordinates);
[ [ Z(7)^0, 0*Z(7) ], [ Z(7)^0, Z(7)^0 ], [ Z(7)^0, Z(7) ],
[ Z(7)^0, Z(7)^2 ], [ Z(7)^0, Z(7)^3 ], [ Z(7)^0, Z(7)^4 ],
[ Z(7)^0, Z(7)^5 ], [ 0*Z(7), Z(7)^0 ] ]
gap> curve := List(coords, t -> VectorSpaceToElement(pg3, [ t[1]^3, t[1]^2 * t[2], t[1] * t[2]^2, t[2]^3 ] ));;
gap> pgl := ProjectivityGroup( pg3 );
The FinInG projectivity group PGL(4,7)
gap> stabcurve := FiningSetwiseStabiliser( pgl, curve );
#I Computing adjusted stabilizer chain...
<projective collineation group with 6 generators>
gap> StructureDescription( stabcurve );
"PSL(3,2) : C2"
gap> Span( curve );
ProjectiveSpace(3, 7)
gap> pg3lines := Lines( pg3 );
<lines of ProjectiveSpace(3, 7)>
gap> orbits := FiningOrbits(stabcurve, pg3lines);
2%..3%..9%..15%..16%..21%..22%..28%..34%..40%..46%..52%..64%..75%..81%..84%..88%..94%..95%..99%..100%..[ <closed orbit, 8 points>, <closed orbit, 56 points>,
<closed orbit, 28 points>, <closed orbit, 168 points>,
<closed orbit, 168 points>, <closed orbit, 28 points>,
<closed orbit, 168 points>, <closed orbit, 28 points>,
<closed orbit, 168 points>, <closed orbit, 168 points>,
<closed orbit, 168 points>, <closed orbit, 168 points>,
<closed orbit, 168 points>, <closed orbit, 336 points>,
<closed orbit, 336 points>, <closed orbit, 168 points>,
<closed orbit, 84 points>, <closed orbit, 112 points>,
<closed orbit, 168 points>, <closed orbit, 21 points>,
<closed orbit, 112 points>, <closed orbit, 21 points> ]
gap> List(orbits, Size);
[ 8, 56, 28, 168, 168, 28, 168, 28, 168, 168, 168, 168, 168, 336, 336, 168,
84, 112, 168, 21, 112, 21 ]
gap> pg3points := Points( pg3 );
<points of ProjectiveSpace(3, 7)>
gap> orbits := FiningOrbits(stabcurve, pg3points);
2%..16%..30%..72%..100%..[ <closed orbit, 8 points>, <closed orbit, 56 points>,
<closed orbit, 56 points>, <closed orbit, 168 points>,
<closed orbit, 112 points> ]
gap> List(orbits, Size);
[ 8, 56, 56, 168, 112 ]
gap> reps := List(orbits, Representative);
[ <a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)>,
<a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)>,
<a point in ProjectiveSpace(3, 7)> ]
gap> x := reps[2];
<a point in ProjectiveSpace(3, 7)>
gap> proj := NaturalProjectionBySubspace(pg3, x);
<geometry morphism from <All elements of ProjectiveSpace(3,
7)> to <All elements of ProjectiveSpace(2, 7)>>
gap> curveminusx := Difference(curve, [x]);
[ <a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)>,
<a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)>,
<a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)>,
<a point in ProjectiveSpace(3, 7)>, <a point in ProjectiveSpace(3, 7)> ]
gap> cuspidal := ImagesSet(proj, List(curveminusx, t -> Span(x, t)));
[ <a point in ProjectiveSpace(2, 7)>, <a point in ProjectiveSpace(2, 7)>,
<a point in ProjectiveSpace(2, 7)>, <a point in ProjectiveSpace(2, 7)>,
<a point in ProjectiveSpace(2, 7)>, <a point in ProjectiveSpace(2, 7)>,
<a point in ProjectiveSpace(2, 7)>, <a point in ProjectiveSpace(2, 7)> ]
gap> coords := List(cuspidal, Coordinates);
[ [ Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ],
[ Z(7)^0, Z(7)^0, Z(7)^0 ], [ Z(7)^0, Z(7)^2, Z(7)^0 ],
[ Z(7)^0, Z(7)^4, Z(7)^0 ], [ Z(7)^0, Z(7)^0, Z(7)^3 ],
[ Z(7)^0, Z(7)^2, Z(7)^3 ], [ Z(7)^0, Z(7)^4, Z(7)^3 ] ]
gap> r := PolynomialRing(GF(7), 3);
GF(7)[x_1,x_2,x_3]
gap> indets := IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3 ]
gap> shapes := Filtered(Tuples([0,1,2,3], 3), t -> Sum(t) = 3);
[ [ 0, 0, 3 ], [ 0, 1, 2 ], [ 0, 2, 1 ], [ 0, 3, 0 ], [ 1, 0, 2 ],
[ 1, 1, 1 ], [ 1, 2, 0 ], [ 2, 0, 1 ], [ 2, 1, 0 ], [ 3, 0, 0 ] ]
gap> mat := List(coords, t -> List(shapes, u -> Product([1,2,3], i -> t[i]^u[i])));
[ [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7),
Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7),
0*Z(7) ],
[ Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0,
Z(7)^0 ],
[ Z(7)^0, Z(7)^2, Z(7)^4, Z(7)^0, Z(7)^0, Z(7)^2, Z(7)^4, Z(7)^0, Z(7)^2,
Z(7)^0 ],
[ Z(7)^0, Z(7)^4, Z(7)^2, Z(7)^0, Z(7)^0, Z(7)^4, Z(7)^2, Z(7)^0, Z(7)^4,
Z(7)^0 ],
[ Z(7)^3, Z(7)^0, Z(7)^3, Z(7)^0, Z(7)^0, Z(7)^3, Z(7)^0, Z(7)^3, Z(7)^0,
Z(7)^0 ],
[ Z(7)^3, Z(7)^2, Z(7), Z(7)^0, Z(7)^0, Z(7)^5, Z(7)^4, Z(7)^3, Z(7)^2,
Z(7)^0 ],
[ Z(7)^3, Z(7)^4, Z(7)^5, Z(7)^0, Z(7)^0, Z(7), Z(7)^2, Z(7)^3, Z(7)^4,
Z(7)^0 ] ]
gap> mat2 := ShallowCopy(mat);
[ [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7),
Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7),
0*Z(7) ],
[ Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0, Z(7)^0,
Z(7)^0 ],
[ Z(7)^0, Z(7)^2, Z(7)^4, Z(7)^0, Z(7)^0, Z(7)^2, Z(7)^4, Z(7)^0, Z(7)^2,
Z(7)^0 ],
[ Z(7)^0, Z(7)^4, Z(7)^2, Z(7)^0, Z(7)^0, Z(7)^4, Z(7)^2, Z(7)^0, Z(7)^4,
Z(7)^0 ],
[ Z(7)^3, Z(7)^0, Z(7)^3, Z(7)^0, Z(7)^0, Z(7)^3, Z(7)^0, Z(7)^3, Z(7)^0,
Z(7)^0 ],
[ Z(7)^3, Z(7)^2, Z(7), Z(7)^0, Z(7)^0, Z(7)^5, Z(7)^4, Z(7)^3, Z(7)^2,
Z(7)^0 ],
[ Z(7)^3, Z(7)^4, Z(7)^5, Z(7)^0, Z(7)^0, Z(7), Z(7)^2, Z(7)^3, Z(7)^4,
Z(7)^0 ] ]
gap> sol := NullspaceMat(TransposedMat(mat2))[1];
[ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^3, Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7),
0*Z(7) ]
gap> terms := List(shapes, u -> Product([1,2,3], i -> indets[i]^u[i]));
[ x_3^3, x_2*x_3^2, x_2^2*x_3, x_2^3, x_1*x_3^2, x_1*x_2*x_3, x_1*x_2^2,
x_1^2*x_3, x_1^2*x_2, x_1^3 ]
gap> poly := terms * sol;
x_1*x_3^2-x_2^3
gap> pg2 := AmbientGeometry(Range(proj));
ProjectiveSpace(2, 7)
gap> variety := ProjectiveVariety(pg2, [poly]);
Projective Variety in ProjectiveSpace(2, 7)
gap> points := Points(variety);
<points of Projective Variety in ProjectiveSpace(2, 7)>
gap> Size(points);
8
[ Dauer der Verarbeitung: 0.4 Sekunden
(vorverarbeitet)
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