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G3SingularProof.m
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/* In this file, we verify the claimed equivalence between quartic singularities
and Dixmier-Ohno relations
This script assumes that the spec file "g3cayley/magma/spec" is loaded
at magma startup (in the .magmarc file)
*/
forward A1NFBiSingular, A1p2NFBiSingular, A2NFBiSingular;
/* The field */
Fld := Rationals();
/* Select the subset of tests you wish */
DOInclusions := true; HNFLargeDims := true; HNFSmallDims := true;
/* 0. A small toolbox
***/
function NormFormSpecialized(f, fctStratum :
VNames := ["a00", "a10", "a01", "a02", "a03", "a04", "a11", "a12", "a13", "a20", "a21", "a22", "a30", "a31", "a40"],
AdditionalConstraints := [],
TestIrreducibility := true, MagmaWorkaround := true)
Fa := BaseRing(Parent(f)); AssignNames(~Fa, VNames);
Pa := RingOfIntegers(Fa); AssignNames(~Pa, VNames);
Pxyz := Parent(f); AssignNames(~Pxyz, ["x", "y", "z"]);
DO := ChangeUniverse(DixmierOhnoInvariants(f : degmax := 21) cat [0], Pa);
//GB := GroebnerBasis(fctStratum(DO));
RD := RadicalDecomposition(ideal<Pa | fctStratum(DO) cat AdditionalConstraints>);
[ Basis(rd) : rd in RD ];
"";
RES := [**];
for rd in RD do
Qa, phi := quo<Pa | rd>; AssignNames(~Qa, VNames);
FQa := FieldOfFractions(Qa); AssignNames(~FQa, VNames);
PXYZ := PolynomialRing(FQa, 3); AssignNames(~PXYZ, ["x", "y", "z"]);
X := PXYZ.1; Y := PXYZ.2; Z := PXYZ.3;
PHI := hom<Pxyz -> PXYZ | X, Y, Z>;
Cfs, Mns := CoefficientsAndMonomials(f);
F := &+[ FQa!phi(Cfs[i])*PHI(Mns[i]) : i in [1..#Cfs]];
"Looking at", F;
if Seqset(DixmierOhnoInvariants(F)) eq {0} then
"\t=> unstable..."; "";
Append(~RES, "unstable");
continue;
end if;
if TestIrreducibility and not IsIrreducible(F) then
FCT := Factorization(F);
"Non irreducible", FCT;
if #FCT eq 1 and FCT[1, 2] eq 2 then
"\t=> c^2..."; "";
Append(~RES, "c^2");
continue;
end if;
end if;
if MagmaWorkaround then
// A workaround to avoid that magma raises an annoying exception
// while computing a radical decomposition over a multivariate quotient ring
// (over finite fields)
xs := Scheme(ProjectiveSpace(Parent(f)),[f]);
sngs := SingularSubscheme(xs);
"Dim Ssing", Dimension(sngs);
PaXYZ := PolynomialRing(CoefficientRing(Pa), Rank(Pa)+3);
AssignNames(~PaXYZ, ["X", "Y", "Z"] cat Names(Pa));
X := PaXYZ.1; Y := PaXYZ.2; Z := PaXYZ.3;
Paphi := hom<Pa -> PaXYZ | [PaXYZ.(i+3) : i in [1..Rank(Pa)]]>;
PAPHI := hom<Pxyz -> PaXYZ | X,Y,Z>;
SNGS := [];
for e in DefiningEquations(sngs) do
Cfs, Mns := CoefficientsAndMonomials(e);
E := &+[ Paphi(Cfs[i])*PAPHI(Mns[i]) : i in [1..#Cfs]];
Append(~SNGS, E);
end for;
RDsngs := RadicalDecomposition(ideal<PaXYZ | SNGS cat [Paphi(e) : e in Basis(rd)]>);//RDsngs;
XS := Scheme(ProjectiveSpace(Parent(F)),[F]);
Types := [];
for rdsngs in RDsngs do
/* We move back rdsngs in FQa... */
RDSNGS := [];
for e in Basis(rdsngs) do
Cfs, Mns := CoefficientsAndMonomials(e);
E := PXYZ!0;
for i->mn in Mns do
exp := Exponents(mn);
E +:= Cfs[i] *
&*[ PXYZ.j^exp[j] : j in [1,2,3] ] *
&*[ FQa.j^exp[3+j] : j in [1..Rank(FQa)] ];
end for;
Append(~RDSNGS, E);
end for;
RDSNGS := GroebnerBasis(RDSNGS);
//RDSNGS;
PTS := [
Evaluate(NormalForm(E, RDSNGS), [FQa|1,1,1]) :
E in [PXYZ.1,PXYZ.2,PXYZ.3]
];
if Seqset(PTS) eq {0} or
Seqset([ Evaluate(E, PTS) : E in DefiningEquations(XS) ]) ne {0} then continue; end if;
pts := XS(FQa)!PTS;
if not IsSingular(pts) then continue; end if;
//Basis(rdsngs),
"->", pts;
boo,F,seq,dat := IsHypersurfaceSingularity(pts,3);
_, _, typ := NormalFormOfHypersurfaceSingularity(F);
Append(~Types, typ);
typ;
end for;
else
/* A more direct way to get these singular points */
P2 := ProjectiveSpace(Parent(F));
XS := Scheme(P2,[F]);
sngs := SingularSubscheme(XS);
"Dim Ssing", Dimension(sngs);
Types := [];
RDsngs := RadicalDecomposition(ideal<PXYZ | DefiningEquations(sngs)>);//RDsngs;
for rdsngs in RDsngs do
//rdsngs;
pts := XS(FQa)![ Evaluate(NormalForm(e, Basis(rdsngs)), [FQa|1,1,1]) : e in [X,Y,Z]];
//Basis(rdsngs),
"->", pts;
boo,F,seq,dat := IsHypersurfaceSingularity(pts,3);
_, _, typ := NormalFormOfHypersurfaceSingularity(F);
Append(~Types, typ);
typ;
end for;
end if; /* end workaround */
"\t=> ", {* t : t in Types *};
"";
Append(~RES, {* t : t in Types *});
end for;
""; "All the possible types are thus", RES;
return RES;
end function;
function SpecialNFinSubStratum(f, fctStratum, SubStratum)
Fa := BaseRing(Parent(f)); Pa := RingOfIntegers(Fa);
DO := ChangeUniverse(DixmierOhnoInvariants(f : degmax := 21) cat [0], Pa);
IdRad := Radical(ideal<Pa | fctStratum(DO)>);
SubEqs := [ NormalForm(E, IdRad) : E in SubStratum(DO) ];
return Seqset(SubEqs) eq {0};
end function;
function IdAdd(id1, id2)
return Seqset(RadicalDecomposition(id1 + id2));
end function;
A2Stratum := func<DO | DixmierOhnoSingularityRelations("(A2)", DO)>;
A1p2Stratum := func<DO | DixmierOhnoSingularityRelations("(A1^2)", DO)>;
A1A2Stratum := func<DO | DixmierOhnoSingularityRelations("(A1A2)", DO)>;
A1p3Stratum := func<DO | DixmierOhnoSingularityRelations("(A1^3)", DO)>;
RedA1p3Stratum := func<DO | DixmierOhnoSingularityRelations("(rA1^3)", DO)>;
RedA1p4cubStratum := func<DO | DixmierOhnoSingularityRelations("(rA1^4_cub)", DO)>;
RedA1p4conStratum := func<DO | DixmierOhnoSingularityRelations("(rA1^4_con)", DO)>;
A2p2Stratum := func<DO | DixmierOhnoSingularityRelations("(A2^2)", DO)>;
A1p2A2Stratum := func<DO | DixmierOhnoSingularityRelations("(A1^2A2)", DO)>;
A1A2p2Stratum := func<DO | DixmierOhnoSingularityRelations("(A1A2^2)", DO)>;
RedA1p3A2Stratum := func<DO | DixmierOhnoSingularityRelations("(rA1^3A2)", DO)>;
RedA1p5Stratum := func<DO | DixmierOhnoSingularityRelations("(rA1^5)", DO)>;
A3Stratum := func<DO | DixmierOhnoSingularityRelations("(A3)", DO)>;
RedA1p6Stratum := func<DO | DixmierOhnoSingularityRelations("(rA1^6)", DO)>;
A2p3Stratum := func<DO | DixmierOhnoSingularityRelations("(A2^3)", DO)>;
RedA1A3Stratum := func<DO | DixmierOhnoSingularityRelations("(rA1A3)", DO)>;
A4Stratum := func<DO | DixmierOhnoSingularityRelations("(A4)", DO)>;
UnstableStratum := func<DO | DO>;
function G3SingularProof(Fld :
DOInclusions := true, HNFLargeDims := true, HNFSmallDims := true);
DomQ := [ Fld |];
AllChecked := true;
/* 1. the DO strata inclusion graph is as awaited?
***/
""; "";
"Invariant criterions for quartic singularities checks, in characteristic p =", Characteristic(Fld);
"---";
""; "";
PR<I27, J21, I21, J18, I18, J15, I15, J12, I12, J09, I09, I06, I03> :=
PolynomialRing(Fld, [27, 21, 21, 18, 18, 15, 15, 12, 12, 09, 09, 06, 03]);
I27, J21, I21, J18, I18, J15, I15, J12, I12, J09, I09, I06, I03 := Explode([PR.i : i in [1..13]]);
DO := [I03,I06,I09,J09,I12,J12,I15,J15,I18,J18,I21,J21,I27];
zeroId := ideal<PR | [
I27, J21, I21, J18, I18, J15, I15, J12, I12, J09, I09, I06, I03 ]>;
DORels := DixmierOhnoAlgebraicRelations(DO);
A2Id := ideal<PR | GroebnerBasis(A2Stratum(DO) cat DORels)>;
A1p2Id := ideal<PR | (A1p2Stratum(DO) cat DORels)>;
A1A2Id := ideal<PR | GroebnerBasis(A1A2Stratum(DO) cat DORels)>;
A1p3Id := ideal<PR | GroebnerBasis(A1p3Stratum(DO) cat DORels)>;
rA1p3Id := ideal<PR | GroebnerBasis(RedA1p3Stratum(DO) cat DORels)>;
rA1p4cubId := ideal<PR | GroebnerBasis(RedA1p4cubStratum(DO) cat DORels)>;
rA1p4conId := ideal<PR | GroebnerBasis(RedA1p4conStratum(DO) cat DORels)>;
A2p2Id := ideal<PR | GroebnerBasis(A2p2Stratum(DO) cat DORels)>;
A1p2A2Id := ideal<PR | GroebnerBasis(A1p2A2Stratum(DO) cat DORels)>;
A1A2p2Id := ideal<PR | GroebnerBasis(A1A2p2Stratum(DO) cat DORels)>;
rA1p3A2Id := ideal<PR | GroebnerBasis(RedA1p3A2Stratum(DO) cat DORels)>;
rA1p5Id := ideal<PR | GroebnerBasis(RedA1p5Stratum(DO) cat DORels)>;
A3Id := ideal<PR | GroebnerBasis(A3Stratum(DO) cat DORels)>;
rA1p6Id := ideal<PR | GroebnerBasis(RedA1p6Stratum(DO) cat DORels)>;
A2p3Id := ideal<PR | GroebnerBasis(A2p3Stratum(DO) cat DORels)>;
rA1A3Id := ideal<PR | GroebnerBasis(RedA1A3Stratum(DO) cat DORels)>;
A4Id := ideal<PR | GroebnerBasis(A4Stratum(DO) cat DORels)>;
if not DOInclusions then
"I skip DO strata inclusions...";
"";
else
"@ DO strata inclusions";
"@@@@@@@@@@@@@@@@@@@@@@";
"";
"Dimension 0 loci";
"================";
"";
"Intersections are as expected?\n";
ret := IdAdd(A4Id, rA1A3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A4>) meet V(<rA1A3>) = V(<unstable>)\t\t\t:", ret;
ret := IdAdd(A4Id, A2p3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A4>) meet V(<A2^3>) = V(<unstable>)\t\t\t:", ret;
ret := IdAdd(A4Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A4>) meet V(<rA1^6>) = V(<unstable>)\t\t\t:", ret;
"";
ret := IdAdd(rA1A3Id, A2p3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1A3>) meet V(<A2^3>) = V(<unstable>)\t\t:", ret;
ret := IdAdd(rA1A3Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1A3>) meet V(<rA1^6>) = V(<unstable>)\t\t:", ret;
"";
ret := IdAdd(A2p3Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A2^3>) meet V(<rA1^6>) = V(<unstable>)\t\t:", ret;
""; "";
"Dimension 1 loci";
"================";
"";
"dim1 -> dim0 inclusions are as expected?\n";
ret := IdAdd(A3Id, A4Id) eq {A4Id}; AllChecked and:= ret;
"\tV(<A3>) meet V(<A4>) = V(<A4>)\t\t\t\t:", ret;
ret := IdAdd(A3Id, rA1A3Id) in { {rA1A3Id}, {zeroId} }; AllChecked and:= ret;
"\tV(<A3>) meet V(<rA1A3>) = V(<rA1A3>), or V(<unstable>)\t:", ret;
ret := IdAdd(A3Id, A2p3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A3>) meet V(<A2^3>) = V(<unstable>)\t\t\t:", ret;
ret := IdAdd(A3Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A3>) meet V(<rA1^6>) = V(<unstable>)\t\t\t:", ret;
"";
ret := IdAdd(A1A2p2Id, A4Id) eq {A4Id}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<A4>) = V(<A4>)\t\t\t:", ret;
ret := IdAdd(A1A2p2Id, rA1A3Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<rA1A3>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(A1A2p2Id, A2p3Id) eq {A2p3Id}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<A2^3>) = V(<A2^3>)\t\t\t:", ret;
ret := IdAdd(A1A2p2Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<rA1^6>) = V(<unstable>)\t\t:", ret;
"";
ret := IdAdd(rA1p3A2Id, A4Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1^3A2>) meet V(<A4>) = V(<unstable>)\t\t:", ret;
ret := IdAdd(rA1p3A2Id, rA1A3Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^3A2>) meet V(<rA1A3>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(rA1p3A2Id, A2p3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1^3A2>) meet V(<A2^3>) = V(<unstable>)\t\t:", ret;
ret := IdAdd(rA1p3A2Id, rA1p6Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1^3A2>) meet V(<rA1^6>) = V(<unstable>)\t\t:", ret;
"";
ret := IdAdd(rA1p5Id, A4Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1^5>) meet V(<A4>) = V(<unstable>)\t\t\t:", ret;
ret := IdAdd(rA1p5Id, rA1A3Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^5>) meet V(<rA1A3>) = V(<rA1A3>)\t\t\t:", ret;
ret := IdAdd(rA1p5Id, A2p3Id) eq {zeroId}; AllChecked and:= ret;
"\tV(<rA1^5>) meet V(<A2^3>) = V(<unstable>)\t\t:", ret;
ret := IdAdd(rA1p5Id, rA1p6Id) eq {rA1p6Id}; AllChecked and:= ret;
"\tV(<rA1^5>) meet V(<rA1^6>) = V(<rA1^6>)\t\t\t:", ret;
"";
"Intersections are as expected?\n";
ret := IdAdd(A3Id, A1A2p2Id) in {{A4Id, rA1A3Id}, {A4Id}} ; AllChecked and:= ret;
"\tV(<A3>) meet V(<A1A2^2>) = {V(<A4>),V(<rA1A3>)}, or V(<A4>):", ret;
ret := IdAdd(A3Id, rA1p3A2Id) in {{rA1A3Id}, {zeroId}}; AllChecked and:= ret;
"\tV(<A3>) meet V(<rA1^3A2>) = V(<rA1A3>), or V(<unstable>):", ret;
ret := IdAdd(A3Id, rA1p5Id) in {{rA1A3Id}, {zeroId}}; AllChecked and:= ret;
"\tV(<A3>) meet V(<rA1^5>) = V(<rA1A3>), or V(<unstable>)\t:", ret;
"";
ret := IdAdd(A1A2p2Id, rA1p3A2Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<rA1^3A2>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(A1A2p2Id, rA1p5Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A1A2^2>) meet V(<rA1^5>) = V(<rA1A3>)\t\t:", ret;
"";
ret := IdAdd(rA1p3A2Id, rA1p5Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^3A2>) meet V(<rA1^5>) = V(<rA1A3>)\t\t:", ret;
""; "";
"Dimension 2 loci";
"================";
"";
"dim2 -> dim1 inclusions are as expected?\n";
ret := IdAdd(A2p2Id, A3Id) eq {A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<A3>) = V(<A3>)\t\t\t:", ret;
ret := IdAdd(A2p2Id, A1A2p2Id) eq {A1A2p2Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<A1A2^2>) = V(<A1A2^2>)\t\t:", ret;
ret := IdAdd(A2p2Id, rA1p3A2Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<rA1^3A2>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(A2p2Id, rA1p5Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<rA1^5>) = V(<rA1A3>)\t\t\t:", ret;
"";
ret := IdAdd(A1p2A2Id, A3Id) eq {A3Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<A3>) = V(<A3>)\t\t\t:", ret;
ret := IdAdd(A1p2A2Id, A1A2p2Id) eq {A1A2p2Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<A1A2^2>) = V(<A1A2^2>)\t\t:", ret;
ret := IdAdd(A1p2A2Id, rA1p3A2Id) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<rA1^3A2>) = V(<rA1^3A2>)\t\t:", ret;
ret := IdAdd(A1p2A2Id, rA1p5Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<rA1^5>) = V(<rA1A3>)\t\t:", ret;
"";
ret := IdAdd(rA1p4conId, A3Id) eq {A3Id}; AllChecked and:= ret;
"\tV(<rA1^4_con>) meet V(<A3>) = V(<A3>)\t\t\t:", ret;
ret := IdAdd(rA1p4conId, A1A2p2Id) eq {rA1A3Id, A4Id}; AllChecked and:= ret;
"\tV(<rA1^4_con>) meet V(<A1A2^2>)={V(<A4>),V(<rA1A3>)}\t:", ret;
ret := IdAdd(rA1p4conId, rA1p3A2Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^4_con>) meet V(<rA1^3A2>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(rA1p4conId, rA1p5Id) eq {rA1p5Id}; AllChecked and:= ret;
"\tV(<rA1^4_con>) meet V(<rA1^5>) = V(<rA1^5>)\t\t:", ret;
"";
"";
ret := IdAdd(rA1p4cubId, A3Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^4_cub>) meet V(<A3>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(rA1p4cubId, A1A2p2Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^4_cub>) meet V(<A1A2^2>) = V(<rA1A3>)\t\t:", ret;
ret := IdAdd(rA1p4cubId, rA1p3A2Id) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<rA1^4_cub>) meet V(<rA1^3A2>) = V(<rA1^3A2>)\t\t:", ret;
ret := IdAdd(rA1p4cubId, rA1p5Id) eq {rA1p5Id}; AllChecked and:= ret;
"\tV(<rA1^4_cub>) meet V(<rA1^5>) = V(<rA1^5>)\t\t:", ret;
"";
"Intersections are as expected?\n";
ret := IdAdd(A2p2Id, A1p2A2Id) eq {A1A2p2Id, A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<A1^2A2>)={V(<A1A2^2>),V(<A3>)}\t:", ret;
ret := IdAdd(A2p2Id, rA1p4conId) eq {A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<rA1^4_con>) = V(<A3>)\t\t\t:", ret;
ret := IdAdd(A2p2Id, rA1p4cubId) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<A2^2>) meet V(<rA1^4_cub>) = V(<rA1A3>)\t\t:", ret;
"";
ret := IdAdd(A1p2A2Id, rA1p4conId) eq {A3Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<rA1^4_con>) = V(<A3>)\t\t:", ret;
ret := IdAdd(A1p2A2Id, rA1p4cubId) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<A1^2A2>) meet V(<rA1^4_cub>) = V(<rA1^3A2>)\t\t:", ret;
"";
ret := IdAdd(rA1p4conId, rA1p4cubId) eq {rA1p5Id}; AllChecked and:= ret;
"\tV(<rA1^4_con>) meet V(<rA1^4_cub>) = V(<rA1^5>)\t\t:", ret;
""; "";
"Dimension 3 loci";
"================";
"";
"dim3 -> dim2 inclusions are as expected?\n";
ret := IdAdd(A1A2Id, A2p2Id) eq {A2p2Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<A2^2>) = V(<A2^2>)\t\t\t:", ret;
ret := IdAdd(A1A2Id, A1p2A2Id) eq {A1p2A2Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<A1^2A2>) = V(<A1^2A2>)\t\t:", ret;
ret := IdAdd(A1A2Id, rA1p4conId) eq {A3Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<rA1^4_con>) = V(<A3>)\t\t\t:", ret;
ret := IdAdd(A1A2Id, rA1p4cubId) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<rA1^4_cub>) = V(<rA1^3A2>)\t\t:", ret;
"";
ret := IdAdd(A1p3Id, A2p2Id) eq {A1A2p2Id, A3Id}; AllChecked and:= ret;
"\tV(<A1^3>) meet V(<A2^2>) = {V(<A1A2^2>), V(<A3>)}\t:", ret;
ret := IdAdd(A1p3Id, A1p2A2Id) eq {A1p2A2Id}; AllChecked and:= ret;
"\tV(<A1^3>) meet V(<A1^2A2>) = V(<A1^2A2>)\t\t:", ret;
ret := IdAdd(A1p3Id, rA1p4conId) eq {rA1p4conId}; AllChecked and:= ret;
"\tV(<A1^3>) meet V(<rA1^4_con>) = V(<rA1^4_con>)\t\t:", ret;
ret := IdAdd(A1p3Id, rA1p4cubId) eq {rA1p4cubId}; AllChecked and:= ret;
"\tV(<A1^3>) meet V(<rA1^4_cub>) = V(<rA1^4_cub>)\t\t:", ret;
"";
ret := IdAdd(rA1p3Id, A2p2Id) eq {rA1A3Id}; AllChecked and:= ret;
"\tV(<rA1^3>) meet V(<A2^2>) = V(<A1A3Id>)\t\t\t:", ret;
ret := IdAdd(rA1p3Id, A1p2A2Id) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<rA1^3>) meet V(<A1^2A2>) = V(<rA1p3A2>)\t\t:", ret;
ret := IdAdd(rA1p3Id, rA1p4conId) eq {rA1p5Id}; AllChecked and:= ret;
"\tV(<rA1^3>) meet V(<rA1^4_con>) = V(<rA1^5>)\t\t:", ret;
ret := IdAdd(rA1p3Id, rA1p4cubId) eq {rA1p4cubId}; AllChecked and:= ret;
"\tV(<rA1^3>) meet V(<rA1^4_cub>) = V(<rA1^4_cub>)\t\t:", ret;
"";
"Intersections are as expected?\n";
ret := IdAdd(A1A2Id, A1p3Id) eq {A1p2A2Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<A1^3>) = V(<A1^2A2>)\t\t\t:", ret;
ret := IdAdd(A1A2Id, rA1p3Id) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<A1A2>) meet V(<rA1^3>) = V(<rA1p3A2>)\t\t:", ret;
"";
ret := IdAdd(A1p3Id, rA1p3Id) eq {rA1p4cubId}; AllChecked and:= ret;
"\tV(<A1^3>) meet V(<rA1^3>) = V(<rA1^4_cub>)\t\t:", ret;
""; "";
"Dimension 4 loci";
"================";
"";
"dim4 -> dim3 inclusions are as expected?\n";
ret := IdAdd(A2Id, A1A2Id) eq {A1A2Id}; AllChecked and:= ret;
"\tV(<A2>) meet V(<A1A2>) = V(<A1A2>)\t\t\t:", ret;
ret := IdAdd(A2Id, A1p3Id) eq {A1p2A2Id}; AllChecked and:= ret;
"\tV(<A2>) meet V(<A1^3>) = V(<A1^2A2>)\t\t\t:", ret;
ret := IdAdd(A2Id, rA1p3Id) eq {rA1p3A2Id}; AllChecked and:= ret;
"\tV(<A2>) meet V(<rA1^3>) = V(<rA1^3A2>)\t\t\t:", ret;
ret := IdAdd(A1p2Id, A1A2Id) eq {A1A2Id}; AllChecked and:= ret;
"\tV(<A1^2>) meet V(<A1A2>) = V(<A1A2>)\t\t\t:", ret;
ret := IdAdd(A1p2Id, A1p3Id) eq {A1p3Id}; AllChecked and:= ret;
"\tV(<A1^2>) meet V(<A1^3>) = V(<A1^3>)\t\t\t:", ret;
ret := IdAdd(A1p2Id, rA1p3Id) eq {rA1p3Id}; AllChecked and:= ret;
"\tV(<A1^2>) meet V(<rA1^3>) = V(<rA1^3>)\t\t\t:", ret;
"";
"Intersections are as expected?\n";
ret := IdAdd(A1p2Id, A2Id) eq {A2Id}; AllChecked and:= ret;
"\tV(<A1^2>) meet V(<A2>) = V(<A2>)\t\t\t:", ret;
""; "";
"Dimension 5 loci";
"================";
"";
"dim5 -> dim4 inclusions are as expected?\n";
"\tV(<A1>) meet V(<A2>) = V(<A2>)\t\t\t\t:", true; /* obviously */
"\tV(<A1>) meet V(<A1^2>) = V(<A1^2>)\t\t\t:", true; /* obviously */
""; "";
end if;
/* 2. The only normal forms with Dixmier-Ohno invariants that satisfy the
stratum equation have the prescribed types?
***/
HuiNF := AssociativeArray();
HuiNF["A1"] := HuiNormalForms("(A1)" : Domain := DomQ) ;
HuiNF["A1^2"] := HuiNormalForms("(A1^2)" : Domain := DomQ) ;
HuiNF["A2"] := HuiNormalForms("(A2)" : Domain := DomQ) ;
HuiNF["rA1^3"] := HuiNormalForms("(rA1^3)" : Domain := DomQ) ;
HuiNF["A1^3"] := HuiNormalForms("(A1^3)" : Domain := DomQ) ;
HuiNF["A1A2"] := HuiNormalForms("(A1A2)" : Domain := DomQ) ;
HuiNF["A3"] := HuiNormalForms("(A3)" : Domain := DomQ) ;
HuiNF["rA1^4_cub"] := HuiNormalForms("(rA1^4_cub)" : Domain := DomQ) ;
HuiNF["rA1^4_con"] := HuiNormalForms("(rA1^4_con)" : Domain := DomQ) ;
HuiNF["A1^2A2"] := HuiNormalForms("(A1^2A2)" : Domain := DomQ) ;
HuiNF["A2^2"] := HuiNormalForms("(A2^2)" : Domain := DomQ) ;
HuiNF["rA1A3"] := HuiNormalForms("(rA1A3)" : Domain := DomQ) ;
HuiNF["A1A3"] := HuiNormalForms("(A1A3)" : Domain := DomQ) ;
HuiNF["A4"] := HuiNormalForms("(A4)" : Domain := DomQ) ;
HuiNF["rA1^5"] := HuiNormalForms("(rA1^5)" : Domain := DomQ) ;
HuiNF["rA1^3A2"] := HuiNormalForms("(rA1^3A2)" : Domain := DomQ) ;
HuiNF["A1A2^2"] := HuiNormalForms("(A1A2^2)" : Domain := DomQ) ;
HuiNF["rA1^2A3a"] := HuiNormalForms("(rA1^2A3_cub)" : Domain := DomQ) ;
HuiNF["rA1^2A3b"] := HuiNormalForms("(rA1^2A3_con)" : Domain := DomQ) ;
HuiNF["rA3^2"] := HuiNormalForms("(rA3^2)" : Domain := DomQ) ;
HuiNF["A2A3"] := HuiNormalForms("(A2A3)" : Domain := DomQ) ;
HuiNF["A1A4"] := HuiNormalForms("(A1A4)" : Domain := DomQ) ;
HuiNF["A5"] := HuiNormalForms("(A5)" : Domain := DomQ) ;
HuiNF["rA1^6"] := HuiNormalForms("(rA1^6)" : Domain := DomQ) ;
HuiNF["A2^3"] := HuiNormalForms("(A2^3)" : Domain := DomQ) ;
HuiNF["rA1^3A3"] := HuiNormalForms("(rA1^3A3)" : Domain := DomQ) ;
HuiNF["rA1A3^2"] := HuiNormalForms("(rA1A3^2)" : Domain := DomQ) ;
HuiNF["rA1A2A3"] := HuiNormalForms("(rA1A2A3)" : Domain := DomQ) ;
HuiNF["rA1A5"] := HuiNormalForms("(rA1A5_con)" : Domain := DomQ) ;
HuiNF["rA7"] := HuiNormalForms("(rA7)" : Domain := DomQ) ;
HuiNF["c^2"] := HuiNormalForms("(c^2)" : Domain := DomQ) ;
HuiNF["A2A4"] := HuiNormalForms("(A2A4)" : Domain := DomQ) ;
HuiNF["A6"] := HuiNormalForms("(A6)" : Domain := DomQ) ;
if not HNFLargeDims then
"I skip specializations for A1, A1^2 & A2...";
"";
else
"@ Specializing A1, A1^2 & A2 normal forms yields one more singularity";
"@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@";
"";
"Stratum A1^2";
"============";
"";
"A1^2 normal forms are in V(<A1^2>)\t:", Seqset(DixmierOhnoSingularityRelations("(A1^2)", DixmierOhnoInvariants(HuiNF["A1^2"]))) eq {0};
""; "";
Pa<a00, a01, a10, a11> := PolynomialRing(Fld, [1, 1, 1, 1]);
PXYZ<X,Y,Z> := PolynomialRing(Pa, 3);
f := a00*Z^4+(X*a10+Y*a01)*Z^3+(X*Y*a11+X^2+Y^2)*Z^2+X^2*Y^2;
DO := DixmierOhnoInvariants(f : degmax := 21) cat [0];
a1p2inv := A1p2NFBiSingular(a00, a01, a10, a11);
for locus in [ "(A4)", "(rA1A3)", "(A2^3)", "(rA1^6)", "(A3)", "(A1A2^2)", "(rA1^3A2)", "(rA1^5)", "(A2^2)", "(A1^2A2)", "(rA1^4_con)", "(rA1^4_cub)", "(A1A2)", "(A1^3)", "(rA1^3)" ] do
printf "A1^2 normal forms in V(<%o>) have one more singularity?\n", locus;
"--------------------";
tt := Cputime(); GB := GroebnerBasis(DixmierOhnoSingularityRelations(locus, DO)); tt := Cputime(tt);
ret := NormalForm(a1p2inv, GB) eq 0; AllChecked and:= ret;
"It turns out that it is", ret;
printf "\t\t\t(checked in %.2os)\n", tt;
"";
end for;
"";
"Stratum A2";
"==========";
"";
"A2 normal forms are in V(<A2>)\t\t:", Seqset(DixmierOhnoSingularityRelations("(A2)", DixmierOhnoInvariants(HuiNF["A2"]))) eq {0};
""; "";
Pa<a02, a03, a11, a12> := PolynomialRing(Fld, [1, 1, 1, 1]);
PXYZ<X,Y,Z> := PolynomialRing(Pa, 3);
f := X^2*Z^2 + X*Y^3 + a12*X*Y^2*Z + a11*X*Y*Z^2 + a03*Y^3*Z + a02*Y^2*Z^2 + Y*Z^3;
DO := DixmierOhnoInvariants(f : degmax := 21) cat [0];
a2inv := A2NFBiSingular(a02, a03, a11, a12);
for locus in [ "(A4)", "(rA1A3)", "(A2^3)", "(A3)", "(A1A2^2)", "(rA1^3A2)", "(A2^2)", "(A1^2A2)", "(A1A2)" ] do
printf "A2 normal forms in V(<%o>) have one more singularity?\n", locus;
"--------------------";
tt := Cputime(); GB := GroebnerBasis(DixmierOhnoSingularityRelations(locus, DO)); tt := Cputime(tt);
ret := NormalForm(a2inv, GB) eq 0; AllChecked and:= ret;
"It turns out that it is", ret;
printf "\t\t\t(checked in %.2os)\n", tt;
"";
end for;
"";
"Stratum A1";
"==========";
"";
"A1 normal forms are in V(<A1>)\t\t:", DixmierOhnoInvariants(HuiNF["A1"])[13] eq 0;
""; "";
//SetVerbose("Groebner", 1);
Pa<a03, a04, a12, a13, a02> := PolynomialRing(Fld, [1, 1, 1, 1, 1]);
PXYZ<X,Y,Z> := PolynomialRing(Pa, 3);
f := Y*Z^3+(Y^2*a02+X^2)*Z^2+(X*Y^2*a12+Y^3*a03+X^2*Y)*Z+a13*X*Y^3+a04*Y^4;
DO := DixmierOhnoInvariants(f : degmax := 21) cat [0];
a1inv := A1NFBiSingular(a03, a04, a12, a13, a02);
for locus in [ "(A4)", "(rA1A3)", "(A2^3)", "(rA1^6)", "(A3)", "(A1A2^2)", "(rA1^3A2)", "(rA1^5)", "(A2^2)", "(A1^2A2)", "(rA1^4_con)", "(rA1^4_cub)", "(A1A2)", "(A1^3)", "(rA1^3)", "(A2)", "(A1^2)" ] do
printf "A1 normal forms in V(<%o>) have one more singularity?\n", locus;
"--------------------";
tt := Cputime(); GB := GroebnerBasis(DixmierOhnoSingularityRelations(locus, DO)); tt := Cputime(tt);
ret := NormalForm(a1inv, GB) eq 0; AllChecked and:= ret;
"It turns out that it is", ret;
printf "\t\t\t(checked in %.2os)\n", tt;
"";
end for;
"";
end if;
if not HNFSmallDims then
"I skip specializations for the small strata...";
"";
else
"@ Specializations of the other normal forms yields the awaited types";
"@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@";
"";
"Stratum A4";
"==========";
"";
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["A4"]))) eq {0}; AllChecked and:= ret;
"A4 normal forms are in V(<A4>)\t\t:", ret;
"";
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["A1A4"]))) eq {0}; AllChecked and:= ret;
"A1A4 normal forms are in V(<A4>)\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["A5"]))) eq {0}; AllChecked and:= ret;
"A5 normal forms are in V(<A4>)\t\t:", ret;
"";
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["rA1A5"]))) eq {0}; AllChecked and:= ret;
"rA1A5 normal forms are in V(<A4>)\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["rA7"]))) eq {0}; AllChecked and:= ret;
"rA7 normal forms are in V(<A4>)\t\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["c^2"]))) eq {0}; AllChecked and:= ret;
"c^2 normal forms are in V(<A4>)\t\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["A2A4"]))) eq {0}; AllChecked and:= ret;
"A2A4 normal forms are in V(<A4>)\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(A4)", DixmierOhnoInvariants(HuiNF["A6"]))) eq {0}; AllChecked and:= ret;
"A6 normal forms are in V(<A4>)\t\t:", ret;
""; "";
"Dimension 1 -> A4";
"*****************";
"";
"A2A3 normal forms in V(<A4>) have at least an A4 singularity?";
"-------------------------------------------------------------";
"";
f := HuiNormalForms("(A2A3)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in { {* "A2", "A4" *} }) or
(Type(T) eq MonStgElt and T in { "unstable" } ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, A2A4 */
"rA3^2 normal forms in V(<A4>) have at least an A4 singularity?";
"--------------------------------------------------------------";
"";
f := HuiNormalForms("(rA3^2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {}) or
(Type(T) eq MonStgElt and T in {"c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"rA1^2A3b normal forms in V(<A4>) have at least an A4 singularity?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^2A3_con)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {}) or
(Type(T) eq MonStgElt and T in {"c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"A1A2^2 normal forms in V(<A4>) have at least an A4 singularity?";
"---------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A2^2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {}) or
(Type(T) eq MonStgElt and T in {"c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"Dimension 2 -> A4";
"*****************";
"";
"A1A3 normal forms in V(<A4>) have at least an A4 singularity?";
"-------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A3)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in { {* "A1", "A4" *} }) or
(Type(T) eq MonStgElt and T in { "unstable" } ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, A1A4 */
"A2^2 normal forms in V(<A4>) have at least an A4 singularity?";
"-------------------------------------------------------------";
"";
f := HuiNormalForms("(A2^2)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, A4Stratum);"";
Tp := SpecialNFinSubStratum(f, A4Stratum, UnstableStratum); AllChecked and:= Tp;
"=> Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"A1^2A2 normal forms in V(<A4>) have at least an A4 singularity?";
"---------------------------------------------------------------";
"";
f := HuiNormalForms("(A1^2A2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {}) or
(Type(T) eq MonStgElt and T in {"c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"rA1^4_con normal forms in V(<A4>) have at least an A4 singularity?";
"---------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^4_con)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {}) or
(Type(T) eq MonStgElt and T in {"c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"Dimension 3 -> A4";
"*****************";
"";
"A3 normal forms in V(<A4>) have at least an A4 singularity?";
"-----------------------------------------------------------";
"";
f := HuiNormalForms("(A3)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in { {* "A4" *} } ) or
(Type(T) eq MonStgElt and T in {"unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, A4 */
"A1A2 normal forms in V(<A4>) have at least an A4 singularity?";
"-------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A2)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, A4Stratum);
Tp := SpecialNFinSubStratum(f, A4Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"A1^3 normal forms in V(<A4>) have at least an A4 singularity?";
"-------------------------------------------------------------";
"";
f := HuiNormalForms("(A1^3)" : Domain := DomQ);
Tp := NormFormSpecialized(f, A4Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {} ) or
(Type(T) eq MonStgElt and T in { "c^2", "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, c^2 */
"";
"Stratum rA1A3";
"=============";
"";
ret := Seqset(DixmierOhnoSingularityRelations("(rA1A3)", DixmierOhnoInvariants(HuiNF["rA1A3"]))) eq {0}; AllChecked and:= ret;
"rA1A3 normal forms are in V(<rA1A3>)\t:", ret;
"";
ret := Seqset(DixmierOhnoSingularityRelations("(rA1A3)", DixmierOhnoInvariants(HuiNF["rA1^2A3a"]))) eq {0}; AllChecked and:= ret;
"rA1^2A3a: normal forms are in V(<rA1A3>)\t:", ret;
"";
ret := Seqset(DixmierOhnoSingularityRelations("(rA1A3)", DixmierOhnoInvariants(HuiNF["rA1^3A3"]))) eq {0}; AllChecked and:= ret;
"rA1^3A3 normal forms are in V(<rA1A3>)\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(rA1A3)", DixmierOhnoInvariants(HuiNF["rA1A3^2"]))) eq {0}; AllChecked and:= ret;
"rA1A3^2 normal forms are in V(<rA1A3>)\t:", ret;
ret := Seqset(DixmierOhnoSingularityRelations("(rA1A3)", DixmierOhnoInvariants(HuiNF["rA1A2A3"]))) eq {0}; AllChecked and:= ret;
"rA1A2A3 normal forms are in V(<rA1A3>)\t:", ret;
""; "";
"Dimension 1 -> rA1A3";
"********************";
"";
"A2A3 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(A2A3)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, RedA1A3Stratum);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"rA3^2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA3^2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
/* OK => unstable, rA1A3^2 */
"rA1^2A3b normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"---------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^2A3_con)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {{* "A1"^^3, "A3" *}} ) or
(Type(T) eq MonStgElt and T in { "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, rA1^3A3 */
"A1A2^2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-------------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A2^2)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, RedA1A3Stratum);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"rA1^3A2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"--------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^3A2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {{* "A1", "A2", "A3" *}} ) or
(Type(T) eq MonStgElt and T in { "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, rA1A2A3 */
"rA1^5 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^5)" : Domain := DomQ);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"Dimension 2 -> rA1A3";
"********************";
"";
"A1A3 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A3)" : Domain := DomQ);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"";
"A2^2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(A2^2)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, RedA1A3Stratum);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;
"";
"Singularities are as expected\t\t:", Tp;
"";
/* OK => unstable */
"A1^2A2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-------------------------------------------------------------------";
"";
f := HuiNormalForms("(A1^2A2)" : Domain := DomQ);
"Unstable:\t", SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
//Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
/* OK => unstable */
"rA1^4_con normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^4_con)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
/* OK => unstable, rA1^3A3 */
"rA1^4_cub normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-------------------------------------------------------------------";
"";
f := HuiNormalForms("(rA1^4_cub)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {{* "A1"^^3, "A3" *}} ) or
(Type(T) eq MonStgElt and T in { "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, A1^3A3 */
"Dimension 3 -> rA1A3";
"********************";
"";
"A1A2 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(A1A2)" : Domain := DomQ);
Tp := NormFormSpecialized(f, RedA1A3Stratum);
"";
ret := &and[ (Type(T) eq SetMulti and T in {{* "A1", "A2", "A3" *}} ) or
(Type(T) eq MonStgElt and T in { "unstable"} ) : T in Tp];
AllChecked and:= ret;
"Singularities are as expected\t\t:", ret;
"";
/* OK => unstable, A1A2A3 */
"A1^3 normal forms in V(<rA1A3>) have at least A1A3 singularities?";
"-----------------------------------------------------------------";
"";
f := HuiNormalForms("(A1^3)" : Domain := DomQ);
//Tp := NormFormSpecialized(f, RedA1A3Stratum);
Tp := SpecialNFinSubStratum(f, RedA1A3Stratum, UnstableStratum); AllChecked and:= Tp;
"Unstable:\t", Tp;