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 SL₃(k). When k is of characteristic 0, Dixmier [Dix87] gave a list of 7 invariants which form a homogeneous system of parameters. It was completed by [Ohn07], who furnished a list of 13 generators for the algebra R(C).

The Dixmier-Ohno invariants are polynomials in the 15 coefficients of a ternary quartic form with coefficients in 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 [LLGR20] 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 I₂₇ of degree 27 plays a particular role. It can be shown that (1)/(2⁴⁰) I₂₇ 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 [BJ14, Def.4.6, Prop.4.7].

The reader may also be interested in [GK06].

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: ∞

    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 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.

> 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 []

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