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A nodal singularity of an algebraic curve is one of the form parameterized by the equation $x y = 0$. A nodal curve is a curve with a nodal singularity.
(e.g. Hain 08, p. 45)
The nodal cubic curve (over some base) is (see at elliptic curve – Nodal curves and cuspidal curves for notation and background) the solution to the Weierstrass equation for which the discriminant vanishes, but the modular invariant $c_4$ does not.
This is equivalently the limit in which the j-invariant $j = \frac{c_4^3}{\Delta}$ goes to $\infty$.
The nodal cubic curve is not an elliptic curve, as it is singular, but adding it to the moduli stack of elliptic curves $\mathcal{M}_{ell}$ produces the compactification $\mathcal{M}_{\overline{ell}}$ which is often relevant.
The formal neighbourhood of the nodal curve in $\mathcal{M}_{\overline{ell}}$ is the Tate curve.
Over the complex numbers, the nodal cubic $E_0$ is the Riemann sphere/complex projective space $\mathbb{P}^1$ with the pole points 0 and $\infty$ identified (hence is a “complex torus with one cycle shrunk away”). Precisely: there is a holomorphic function
which is onto $E_0 \subset \mathcal{P}_2$, sends the unit of the multiplicative group $1 \in \mathbb{C}^\times \hookrightarrow \mathbb{P}^1$ to the unit of $E_0$, maps $0,\infty \in \mathbb{P}^1$ both to the nodal singular double point of $E_0$ and is injective away from these points (e.g. Hain 08, exercise 47, p. 45)
The formal group associated with a nodal cubic curve is of height 1. Indeed, passing to the point of the nodal curve in $\mathcal{M}_{\overline{ell}}$ connects elliptic cohomology (of chromatic level 2) to topological K-theory (of chromatic level 1). For more on this see at moduli stack of tori and at tmf – Properties – Maps to K-theory and to Tate K-theory.
In F-theory the nodal singularity locus of the given elliptic fibration is interpreted as the locus of D7-branes, see at F-brane scan.
in F-theory the points where the fibers of the elliptic fibration degenerate to the nodal curve are where the D7-branes are located
Discussion over the complex numbers is in
Last revised on December 21, 2020 at 07:17:04. See the history of this page for a list of all contributions to it.