higher geometry / derived geometry
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For a scheme, a cubic curve over is a scheme over equipped with a section and such that Zariski locally on , is given by a solution in projective space of an equation of the form
(the Weierstrass equation) such that is the line at infinity.
Equivalently this says that 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 for the moduli stack of such cubic curves. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant
(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) | ||||||||
structure group of covering | (modular group) | ||||||||
moduli stack | (M_ell) | (M_fg) | |||||||
of | 1d tori | Tate curves | elliptic curves | cubic curves | 1d commutative formal groups | ||||
value of structure sheaf over curve | KU | elliptic spectrum | complex oriented cohomology theory | ||||||
spectrum of global sections of structure sheaf | (KO KU) = KR-theory | Tate K-theory () | (Tmf Tmf(n)) (modular equivariant elliptic cohomology) | tmf |
There is an eight-fold cover of localized at (Mathew 13, section 4.2) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the -action on KU whose homotopy fixed points are KO).
Reviews for the case that 2 and 3 are invertible include
and specifically over the complex numbers:
Discussion of the general case in the context of the construction of tmf is in
reviewed in
Last revised on June 4, 2020 at 13:58:42. See the history of this page for a list of all contributions to it.