Invariants#

Isomorphisms of plane smooth quartics over an algebraically closed field \(k\) are induced by linear transformations of the ambient projective plane. Therefore, isomorphism classes are characterized by the weighted projective space associated to the ring of invariants of ternary quartic forms under the classical action of \({\rm SL}_3(k)\). When \(k\) is of characteristic \(0\), Dixmier [Dixmier, 1987] gave a list of \(7\) invariants which form a homogeneous system of parameters. It was completed by [Ohno, 2007], who furnished a list of \(13\) generators for the algebra \(R({\mathbb{C}})\).

The Dixmier-Ohno invariants are polynomials in the \(15\) coefficients of a ternary quartic form with coefficients in \({\mathbb{Z}}[1/6]\). They can be considered as a point in the weighted projective space with weights \((3,6,9,9,12,12,15,15,18,18,21,21,27)\). A list of generators of the invariants of smooth plane quartics in positive characteristic is not known, although it is suspected that the reduction of the Dixmier–Ohno invariants are generators when the characteristic is greater than \(7\). In [Lercier et al., 2020] homogeneous systems of parameters are determined in all characteristics except \(3\), for which there is a conjectural HSOP that involves an invariant of degree \(81\). A call to DixmierOhnoInvariants() in general characteristic outputs a minimal set of invariants that generate the largest subring of invariants that is known so far.

Among the Dixmier–Ohno invariants of a form \(f(x,y,z)\), the invariant \(I_{27}\) of degree \(27\) plays a particular role. It can be shown that \({1}\over{2^{40}} I_{27}\) has integral coefficients: it defines the so-called discriminant of ternary quartic forms. Over any field, the zero locus of the discriminant is precisely the set of ternary quartic forms that define a singular plane quartic. The current calculation of this discriminant is based on the techniques developed in [Busé and Jouanolou, 2014] (Def.4.6, Prop.4.7).

The reader may also be interested in [Girard and Kohel, 2006].

DixmierOhnoInvariants(C): Crv -> SeqEnum, SeqEnum#

DixmierOhnoInvariants(f): RngMPolElt -> SeqEnum, SeqEnum#

normalize            : BoolElt                      Default: false
IntegralNormalization: BoolElt                      Default: false
PrimaryOnly          : BoolElt                      Default: false
degmin               : RngIntElt                    Default: 1
degmax               : RngIntElt                    Default: Infinity()
PolynomialOnly       : BoolElt                      Default: true

Compute the Dixmier-Ohno invariants of a ternary quartic form \(f\). When the characteristic of the coefficient ring is 0 or greater than 7, the returned invariants are ‘I3’, ‘I6’, ‘I9’, ‘J9’, ‘I12’, ‘J12’, ‘I15’, ‘J15’, ‘I18’, ‘J18’, ‘I21’, ‘J21’ and ‘I27’. Weights of these invariants are returned as well. If normalize is set to true, then the invariants are normalized in the corresponding weighted projective space before being returned.

Setting IntegralNormalization to true multiplies the Dixmier-Ohno invariants by certain constants so that the invariants (as polynomials in the coefficients) are defined over \({\mathbb{Z}}\). Using the flags degmin and degmax provides only a partial list of generators in the corresponding degrees. Setting the flag PolynomialOnly to false (only relevant in characteristic \(3\)) provides additional invariants that come from non-integral expressions in the Dixmier–Ohno invariants.

Example: Dixmierohno Ex (ex-11474d)#

We calculate the Dixmier-Ohno invariants of the Klein quartic and one of its non-trivial twists over \({\mathbb{Q}}\).

> P<x,y,z> := PolynomialRing(Rationals(), 3);
> PP := ProjectiveSpace(P);
> f1 := x^3*y + y^3*z + z^3*x;
> f2 := x^4 + 7*x^3*z + 3*x^2*y^2 - 3*x^2*z^2 - 6*x*y*z^2 - 5*x*z^3 +
> 2*y^3*z + 3*y^2*z^2 + 2*y*z^3 - 4*z^4;
> C1 := Curve(PP, f1); DO1 := DixmierOhnoInvariants(C1 : normalize := true);
> C2 := Curve(PP, f2); DO2 := DixmierOhnoInvariants(C2 : normalize := true);
> DO1 eq DO2;
true
> IsIsomorphicPlaneQuartics(C1, C2);
false []

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DiscriminantOfTernaryQuartic(f): RngMPolElt -> Any#: Compute the discriminant of the ternary quartic form \(f\).

Example: Discriminant Ex (ex-a1e8bd)#

We compute the discriminant of the Klein Quartic over \({\mathbb{F}}_2\).

> P<x,y,z> := PolynomialRing(GF(2), 3);
> Q := x^3*y + y^3*z + z^3*x;
> DiscriminantOfTernaryQuartic(Q);
1

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DiscriminantFromDixmierOhnoInvariants(DO): SeqEnum -> Any#: Compute the discriminant of a ternary quartic form from the given Dixmier-Ohno invariants \(DO\).

DixmierOhnoInvariantsEqual(DO1, DO2): SeqEnum, SeqEnum -> BoolElt#: Check whether Dixmier-Ohno Invariants \(DO1\) and \(DO2\) of two quartics are equivalent.

DixmierOhnoAlgebraicRelations(DOinv): SeqEnum -> SeqEnum#: Return generators of the ideal of relations between the Dixmier-Ohno invariants.

CovariantHessian(Phi): RngMPolElt -> RngMPolElt#: Compute the Hessian covariant of the ternary quartic form \(\Phi\).

ContravariantSigmaAndPsi(Phi): RngMPolElt -> RngMPolElt, RngMPolElt#: Compute the covariants \(\Sigma\) and \(\Psi\) of a ternary quartic form \(\Phi\), as defined in [Salmon, 1879, p. 78].

QuarticCovariantsAndContravariants(Phi): RngMPolElt -> SeqEnum#: Computes generators of the covariant and contravariant algebra of the ternary quartic form \(\Phi\).