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Commit 2af1f8cb authored by Luke Naylor's avatar Luke Naylor
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Define beta +- in zero/positive rank cases

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......@@ -356,11 +356,27 @@ degenerate_characteristic_curves
\label{fig:charact_curves_vis}
\end{figure}
\begin{definition}[$\beta_{\pm}$]
\label{dfn:beta_pm}
Given a formal Chern character $v$ with positive rank, we define $\beta_{\pm}(v)$ to be
the $\beta$-coordinate of where $\Theta_v^{\pm}$ meets the $\beta$-axis:
\[
\beta_\pm(R,C\ell,D\ell^2) = \frac{C \pm \sqrt{C^2-2RD}}{R}
\]
\noindent
In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
equation $\chern_2^{\beta}(v)=0$.
This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
\end{definition}
\subsubsection{Rank Zero Case}
\begin{sagesilent}
from rank_zero_case import Theta_v_plot
\end{sagesilent}
\subsubsection{Rank Zero Case}
\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
......@@ -387,7 +403,7 @@ Indeed:
\begin{align*}
\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
\\
\text{as} \: 0<\varepsilon &\longrightarrow 0
\text{as} \:\: 0<\varepsilon &\longrightarrow 0
\end{align*}
So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
But also, considering the base point of $\Theta_v^-$:
......@@ -404,6 +420,19 @@ For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
as $\Theta_v^-$ to be able to use the same terminology in both positive rank
and rank zero cases.
\begin{definition}[Extending $\beta_-$ to rank 0 case]
\label{dfn:beta_-_rank0}
Given a formal Chern character $v$ with rank 0 and $\chern_1(v)>0$, we define
$\beta_-(v)$ to be the $\beta$-coordinate of point where $\Theta_v$ meets the
$\beta$-axis:
\[
\beta_-(0,C\ell,D\ell^2) = \frac{D}{C}
\]
\noindent
If $\beta_+$ were also to be generalised to the rank 0 case, we would consider
its value to be $+\infty$ due to the discussion above.
\end{definition}
\subsection{Relevance of $V_v$}
\label{subsect:relevance-of-V_v}
......
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