| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/fining/examples/include/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 27.6.2023 mit Größe 6 kB |
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Quelle examples_varieties.include Sprache: unbekannt |
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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 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% <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: 0.15 Sekunden (vorverarbeitet) ] |
2026-03-28