cubic curve in nLab

Contents

Contents

Definition

For SS a scheme, a cubic curve over SS is a scheme p:X→Sp \colon X \to S over SS equipped with a section e:S→Xe \colon S \to X and such that Zariski locally on SS, XX is given by a solution in projective space ℙ S 2\mathbb{P}_S^2 of an equation of the form

y 2+a 1xy=x 3+a 2x 2+a 4x+a 6 y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6

(the Weierstrass equation) such that e:S→Xe \colon S \to X is the line at infinity.

Equivalently this says that pp is a proper flat morphism with a section contained in the smooth locus whose fibers are geometrically integral curves of arithmetic genus one.

A non-singular solution to this equation is an elliptic curve (see there for more). Write ℳ cub\mathcal{M}_{cub} for the moduli stack of such cubic curves. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant Δ∈H 0(ℳ cub,ω 12)\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})

(e.g. Mathew, section 3)

Properties

Covers

There is an eight-fold cover of ℳ cub\mathcal{M}_{cub} localized at 22 (Mathew 13, section 4.2) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the ℤ 2\mathbb{Z}_2-action on KU whose homotopy fixed points are KO).

References

Reviews for the case that 2 and 3 are invertible include

• Balázs Szendrői, Cubic curves: a short survey (pdf)

and specifically over the complex numbers:

• Richard Hain, section 5 of Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)

Discussion of the general case in the context of the construction of tmf is in

• Akhil Mathew, The homology of tmftmf (arXiv:1305.6100)

reviewed in

• Akhil Mathew, section 3 of: The homotopy groups of TMFTMF (pdf)