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\begin{document}

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\section*{nodal curve}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_nodal_cubic}{The nodal cubic}\dotfill \pageref*{the_nodal_cubic} \linebreak
\noindent\hyperlink{properties_of_the_nodal_cubic}{Properties of the nodal cubic}\dotfill \pageref*{properties_of_the_nodal_cubic} \linebreak
\noindent\hyperlink{compactified_moduli_stack_of_elliptic_curves_and_the_tate_curve}{Compactified moduli stack of elliptic curves and the Tate curve}\dotfill \pageref*{compactified_moduli_stack_of_elliptic_curves_and_the_tate_curve} \linebreak
\noindent\hyperlink{OverTheComplexNumbers}{Over the complex numbers}\dotfill \pageref*{OverTheComplexNumbers} \linebreak
\noindent\hyperlink{formal_group_and_height}{Formal group and height}\dotfill \pageref*{formal_group_and_height} \linebreak
\noindent\hyperlink{relation_to_gauge_enhancement_in_ftheory}{Relation to gauge enhancement in F-theory}\dotfill \pageref*{relation_to_gauge_enhancement_in_ftheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{nodal} [[singular point of an algebraic variety|singularity]] of an [[algebraic curve]] is one of the form parameterized by the [[equation]] $x y = 0$. A \emph{nodal curve} is a curve with a nodal singularity.

(e.g.\hyperlink{Hain08}{Hain 08, p. 45})

\hypertarget{the_nodal_cubic}{}\subsection*{{The nodal cubic}}\label{the_nodal_cubic}

For nodal [[cubic curve]] (over some base) is (see at \emph{\href{elliptic+curve#EllipticCurvesNodalCurvesCuspidalCurves}{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.

Notice that this means equivalently that this is the limit in which the [[j-invariant]] $j = \frac{c_4^3}{\Delta}$ goes to $\infty$.

\hypertarget{properties_of_the_nodal_cubic}{}\subsection*{{Properties of the nodal cubic}}\label{properties_of_the_nodal_cubic}

\hypertarget{compactified_moduli_stack_of_elliptic_curves_and_the_tate_curve}{}\subsubsection*{{Compactified moduli stack of elliptic curves and the Tate curve}}\label{compactified_moduli_stack_of_elliptic_curves_and_the_tate_curve}

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 [[Deligne-Mumford compactification|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]].

\hypertarget{OverTheComplexNumbers}{}\subsubsection*{{Over the complex numbers}}\label{OverTheComplexNumbers}

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 manifold|complex]] [[torus]] with one cycle shrunk away''). Precisely: there is a [[holomorphic function]]

\begin{displaymath}
\mathcal{P}^1 \to \mathcal{P}^2
\end{displaymath}
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. \hyperlink{Hain08}{Hain 08, exercise 47, p. 45})

\begin{displaymath}
\itexarray{
     \mathbb{C}^\times &\hookrightarrow& E_0
     \\
     \downarrow^\mathrlap{=} && \downarrow
     \\
     \mathbb{P}^1-\{0,\infty\} &\longrightarrow& \mathbb{P}^2
  }
  \,.
\end{displaymath}


\hypertarget{formal_group_and_height}{}\subsubsection*{{Formal group and height}}\label{formal_group_and_height}

The [[formal group]] associated with a nodal cubic curve is of [[height of a formal group|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 \emph{[[moduli stack of tori]]} and at \emph{\href{tmf#MapToTateKTheory}{tmf -- Properties -- Maps to K-theory and to Tate K-theory}}.

\hypertarget{relation_to_gauge_enhancement_in_ftheory}{}\subsubsection*{{Relation to gauge enhancement in F-theory}}\label{relation_to_gauge_enhancement_in_ftheory}

In [[F-theory]] the nodal singularity locus of the given [[elliptic fibration]] is interpreted as the locus of [[D7-branes]], see at \emph{\href{F-theory#Fbranescan}{F-brane scan}}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cusp]]


\item [[augmented Teichmüller space]]


\item in [[F-theory]] the points where the fibers of the [[elliptic fibration]] degenerate to the nodal curve are where the [[D7-branes]] are located



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion over the [[complex numbers]] is in

\begin{itemize}%
\item Richard Hain, section 5.2 of \emph{Lectures on Moduli Spaces of Elliptic Curves} (\href{http://arxiv.org/abs/0812.1803}{arXiv:0812.1803})

\end{itemize}
[[!redirects nodal curves]]

[[!redirects nodal cubic curve]] [[!redirects nodal cubic curves]]

[[!redirects nodal cubic]] [[!redirects nodal cubics]]

[[!redirects nodal singularity]] [[!redirects nodal singularities]]



\end{document}