Creation from Invariants

Given the Dixmier-Ohno invariants of a generic plane smooth quartic over a field k of characteristic 0, the algorithms developed in [LRS16] allow the reconstruction of a model of this quartic, which is returned over k itself as long as the geometric automorphism group of C is not of order 2. The main function to this end is PlaneQuarticFromDixmierOhnoInvariants. It is proven to work only in characteristic 0 although it is likely to work in characteristic ≥11 as well (and one can check if the result is correct anyway).

In the above, "generic" means concretely that the Dixmier-Ohno invariant I₁₂ is different from 0. If I₁₂ is zero, other systems of co- or contra-variants may be chosen to perform the reconstruction. These variants have not been implemented, and there are smooth plane quartics, like the Klein quartic, for which no such system exists. Regardless, for all non-trivial automorphism strata except for the cyclic group Z/2Z, as well as for (Z/2Z)² in case I₁₂ = 0, an ad hoc reconstruction is performed.

If the quartic curve has automorphism group of order 2, the field of moduli is not necessarily a field of definition and the reconstruction may happen over a quadratic extension only. Still, the algorithms will in practice often find a model over the field of moduli if it exists.

For more details, see [LRS18], [Els15].

PlaneQuarticFromDixmierOhnoInvariants(DO) : SeqEnum -> Crv, SeqEnum

TernaryQuarticFromDixmierOhnoInvariants(DO) : SeqEnum -> RngMPolElt, SeqEnum

    exact: BoolElt                      Default: false

    minimize: BoolElt                   Default: true

    descent: BoolElt                    Default: true

    search_point: BoolElt               Default: true

Reconstructs a plane quartic from a given tuple of Dixmier-Ohno invariants DO.

If the flag exact is set to true, then a ternary forms is returned whose Dixmier-Ohno invariants exactly equal DO (instead of merely being equal in the corresponding weighted projective space).

If the flag descent is set to true, then the curve is descended to its base field.

If the flag minimize is set to true, then over the rationals an effort is made to return as small a model as possible.

If the flag search_point is set to true, then the algorithm tries to find a rational point on the Mestre conic of the associated binary form. This is required when reconstructing over the base field.

> P<x,y,z> := PolynomialRing(GF(31), 3);
> PP := ProjectiveSpace(P);
> f1 := x^4 + 3*y^4 + 5*z^4 + x^2*y*z + x*y*z^2 + x^2*y^2;
> C1 := Curve(PP, f1);
> I := DixmierOhnoInvariants(f1);
> C2 := Curve(PP, TernaryQuarticFromDixmierOhnoInvariants(I));
> IsIsomorphicPlaneQuartics(C1, C2);
true [
    [ 1 24  8]
    [ 8 27 20]
    [13 20 19]
]