From d3629091819687fcba930cfa6a4505257ea0b64f Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 1 Jun 2023 12:27:14 +0100
Subject: [PATCH] Remove 's for: is
---
main.tex | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
index f279fad..8b19ac7 100644
--- a/main.tex
+++ b/main.tex
@@ -79,7 +79,7 @@ There are some Bogomolov-Gieseker type inequalities:
$0 \leq \Delta(E), \Delta(G)$ and $\Delta(E) + \Delta(G) \leq \Delta(F)$.
We also have a condition relating to the tilt category $\firsttilt\beta$:
$0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
-Finally, there's a condition ensuring that the radius of the circular wall is
+Finally, there is a condition ensuring that the radius of the circular wall is
strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.
For any fixed $\chern_0(E)$, the inequality
@@ -383,7 +383,7 @@ The constant term in $r$ is
$\chern^{\beta}_2(F)/2 + \beta q$.
The linear term in $r$ is
$\sage{bgmlv1_d_lowerbound_linear_term}$.
-Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
+Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
$\frac{R\chern^{\beta}_2(F)/2 + R\beta q - Cq + q^2 }{2r-R}$.
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
@@ -470,7 +470,7 @@ bgmlv2_d_upperbound_exp_term = (
\end{sagesilent}
Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
-of $r$ again, there's a constant term
+of $r$ again, there is a constant term
$\sage{bgmlv2_d_upperbound_const_term}$,
a linear term
$\sage{bgmlv2_d_upperbound_linear_term}$,
@@ -631,7 +631,7 @@ The constant term in $r$ is
$\chern^{\beta}_2(F) + \beta q$.
The linear term in $r$ is
$\sage{bgmlv3_d_upperbound_linear_term}$.
-Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
+Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
\bgroup
\def\psi{\chern_1^{\beta}(F)}
@@ -833,7 +833,7 @@ def plot_d_bound(
\end{figure}
Recalling that $q := \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
-it's worth noting that the extreme values of $q$ in this range lead to the
+it is worth noting that the extreme values of $q$ in this range lead to the
tightest bounds on $d$, as illustrated in figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
@@ -842,7 +842,7 @@ $\chern_0(E)=:r>R:=\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
(eqn \ref{eqn:positive_rad_d_bound_betamin}) match when $q=0$.
-In the other case, $q=\chern^{\beta}_1(F)$, it's the right hand sides of
+In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
(eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
(eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
--
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