nLab cubic curve

Definition

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

$$y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6$$

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

Equivalently this says that $p$ 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 $\mathcal{M}_{cub}$ for the moduli stack of such cubic curves. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$

(e.g. Mathew, section 3)

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering					by of level-n structures (modular curve)
	$\ast = Spec(\mathbb{Z})$	$\to$	$Spec(\mathbb{Z}[ [q] ])$	$\to$	$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering	$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack	$\mathcal{M}_{1dTori}$	$\hookrightarrow$	$\mathcal{M}_{Tate}$	$\hookrightarrow$	$\mathcal{M}_{\overline{ell}}$ (M_ell)	$\hookrightarrow$	$\mathcal{M}_{cub}$	$\to$	$\mathcal{M}_{fg}$ (M_fg)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)		(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)		tmf		$\mathbb{S}$

Properties

Covers

There is an eight-fold cover of $\mathcal{M}_{cub}$ localized at $2$ (Mathew 13, section 4.2) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the $\mathbb{Z}_2$-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 $tmf$ (arXiv:1305.6100)

reviewed in

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