Cubic form

In

zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve

In (

action on the space of integral binary cubic forms and cubic rings up to isomorphism

The classification of real cubic forms $ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}$ is linked to the classification of umbilical points of surfaces. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus.^[2]

Examples

Notes

^ In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.
ISBN 978-0-521-00264-6

References

MR 0160744

Gan, Wee-Teck;
MR 1932327

Iskovskikh, V.A.;
Popov, V.L. (2001) [1994], "Cubic form", Encyclopedia of Mathematics, EMS Press

Iskovskikh, V.A.;
Popov, V.L. (2001) [1994], "Cubic hypersurface", Encyclopedia of Mathematics, EMS Press

MR 0833513

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[1] In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.

[port-2] ISBN 978-0-521-00264-6

[2]