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// This is a comment. | ||
// This is another comment. | ||
public void thing() { | ||
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} |
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This is a test. This is a test. This is a test. This is a test. % This is a comment | ||
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% Must not omit this very important statement: | ||
Very important statement | ||
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We refer the reader to the proof in Lemma~\ref{lem:FLT} | ||
% Here is some text | ||
% that I have decided | ||
% to remove. | ||
the proof given is particularly elegant because of reasons X,Y,Z... | ||
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% better for a higher number of surfaces. | ||
The run-time of the DT method does not depend on the number of surfaces and | ||
scales well even for a high number of surfaces. | ||
% | ||
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\documentclass[12pt]{article} | ||
\usepackage{amsmath} | ||
\title{\LaTeX} | ||
\date{} | ||
\begin{document} | ||
\maketitle | ||
\LaTeX{} is a document preparation system for the \TeX{} | ||
typesetting program. It offers programmable desktop | ||
publishing features and extensive facilities for | ||
automating most aspects of typesetting and desktop | ||
publishing, including numbering and cross-referencing, | ||
tables and figures, page layout, bibliographies, and | ||
much more. \LaTeX{} was originally written in 1984 by | ||
Leslie Lamport and has become the dominant method for | ||
using \TeX; few people write in plain \TeX{} anymore. | ||
The current version is \LaTeXe. | ||
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% This is a comment, not shown in final output. | ||
% The following shows typesetting power of LaTeX: | ||
\begin{align} | ||
E_0 &= mc^2 \\ | ||
E &= \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} | ||
\end{align} | ||
\end{document} | ||
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\begin{itemize} | ||
\item $\pi \colon Bl_J M \rightarrow M$ is the blowup along | ||
the Jacobian ideal $J = (Jac(f), f)$ for $X$. Note that if we | ||
wanted to we could have instead used | ||
$J = (Jac(f,g), f, g) \cdot (Jac(f), f)$; this would give | ||
us a blowup that factored through both the | ||
Nash blowup of $Z$, and the blowup that gives us Aluffi's | ||
formula (which holds under further blowups). $\X$, ${\cal H}$ | ||
denote the total transforms of $X$ and $H$ under the blowup $Bl_J M$, | ||
and $E_X$ denotes the exceptional divisor over $X$. % actually E_X = E_Y = E_Z = E? or not | ||
jfkldskjfadlksajfklds | ||
\item $i_{X \cap H}^* \pi_* = \pi_* i_{\X \cap {\cal H}}^*$, | ||
where $i_{\X \cap {\cal H}}^*$ is the transverse intersection map | ||
for $\X \cap {\cal H}$. | ||
\item $i_{\X \cap {\cal H}}^*[\X]$ is equal to | ||
$[\X \pitch {\cal H}]$, and this is the same class as $[\Z]$; | ||
similarly for $i_{\X \cap {\cal H}}^*[E_X]$. | ||
\end{itemize} | ||
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Let $X$ and $Y$ be smooth hypersurfaces in a smooth ambient variety $M$, | ||
where $Z = X \cap Y$ is a smooth global complete intersection and | ||
$X$ meets $Y$ transversely. If $\nu_Z$, $\nu_X$, $\nu_Y$ denote the | ||
normal bundles of $Z$, $X$, and $Y$ in $M$, then the transversality of | ||
$X$ and $Y$ means that $\nu_Z = \nu_X|_Z \oplus \nu_Y|_Z$. | ||
Moreover, the Chern class $c(\nu_Z)$ is given by $1 + Z$, where $Z$ | ||
denotes the dual of the fundamental class $[Z]$, {\it i.e.} | ||
$Z \cap [M] = [Z]$ (and similarly for $c(\nu_X)$ and $c(\nu_Y)$). % missing inclusions? | ||
Using these facts we can compute the following formula for the | ||
MacPherson-Chern class $c_*(Z)$ of $Z$ in terms of $X$ and $Y$: | ||
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Lorem ipsum dolor sit amet, consectetur adipisicing elit, % Comment asdf fdsa alskdjf dsfaj fkdljsafkdsj aflkdj salfk jdslkafj jd salkfjd. | ||
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Lorem ipsum dolor sit amet, consectetur adipisicing elit, % Comment asdf fdsa | ||
% alskdjf dsfaj | ||
% fkdljsafkdsj aflkdj | ||
% salfk jdslkafj jd | ||
% salkfjd. |