Xiaolong Liu
Xiaolong Liu's Blog
I'm interested in higher dimensional algebraic geometry, birational geometry, moduli space of varieties and bundles and derived categories of sheaves.

Some of the rules of typing blogs.

~

Some basic rules and codes in this Jekyll and HTML.

Inline HTML elements

HTML defines a long list of available inline tags, a complete list of which can be found on the Mozilla Developer Network.

  • To bold text, use **To bold text**.
  • To italicize text, use *To italicize text*.
  • To highlight, use <mark>To highlight</mark>.
  • Abbreviations, like HTML should use <abbr>, with an optional title attribute for the full phrase.
  • Citations, like — Mark Otto, should use <cite>.
  • Deleted text should use <del> and inserted text should use <ins>.
  • Superscript text uses <sup> and subscript text uses <sub>.

Most of these elements are styled by browsers with few modifications on our part.

How to use LaTeX here?

Basic settings

First, we add the following codes in _includes/head.html:

<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<script type="text/x-mathjax-config">

MathJax.Ajax.config.path["Contrib"] = "//cdn.mathjax.org/mathjax/contrib";

  MathJax.Hub.Config({
    tex2jax: {
      skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'],
      inlineMath: [              // start/end delimiter pairs for in-line math
      ['$', '$']
    ],
    displayMath: [             // start/end delimiter pairs for display math
      ['$$', '$$'],
      ['\\[', '\\]']
    ],
    processEscapes: true,
    processEnvironments: true,
    },
    
    TeX: { 
    equationNumbers: {autoNumber: "AMS"},
    extensions: ["[Contrib]/xyjax/xypic.js"]
    }
  });
</script>

After set math: true, we can use the LaTeX codes, such as $\int_{S}\omega\in{\iint}$ given by

$\int_{S}\omega\in\{\iint\}$

and \[ H^i_{DR}(X/S):=H^iR\Gamma(X,\Omega_{X/S}^*) \] given by

\\[
H^i_{DR}(X/S):=H^iR\Gamma(X,\Omega_{X/S}^*)
\\]

Hence we can write math now!

Note that we need to use \_ instead of _ and use \\{…\\} instead of \{…\}.

Math environments and mutiple lines

When $a \ne 0$, there are two solutions to $ax^2 + bx + c = 0$ and they are \begin{align} DDD\\ eee\\ eee \end{align} and \begin{align} &\left\langle L_1,M\right\rangle _{\pi}\otimes\left\langle L_2,M\right\rangle _{\pi}\cong\left\langle L_1\otimes L_2,M\right\rangle _{\pi}\\ &\left\langle L,M_1\right\rangle _{\pi}\otimes\left\langle L,M_2\right\rangle _{\pi}\cong\left\langle L,M_1\otimes M_2\right\rangle _{\pi}; \end{align}

given by

<html>
<head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width">
  <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
  <script id="MathJax-script" async
          src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
  </script>
</head>
<body>
<p>
  When $a \ne 0$, there are two solutions to $ax^2 + bx + c = 0$ and they are
  \begin{align}
  DDD\\
  eee\\
  eee
  \end{align}
and
\begin{align}
&\left\langle L_1,M\right\rangle _{\pi}\otimes\left\langle L_2,M\right\rangle _{\pi}\cong\left\langle L_1\otimes L_2,M\right\rangle _{\pi}\\
&\left\langle L,M_1\right\rangle _{\pi}\otimes\left\langle L,M_2\right\rangle _{\pi}\cong\left\langle L,M_1\otimes M_2\right\rangle _{\pi};
\end{align}
</p>
</body>
</html>

Commutative diagrams I - AMSCD environment (not useful)

As

\begin{equation} \require{AMScd} \begin{CD} S' @>^{66} >> Y\\ @V^{55}VV @VV V\\ S @> >> X \end{CD} \end{equation}

given by

<html>
<head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width">
  <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
  <script id="MathJax-script" async
          src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
  </script>
</head>
<body>
<p>
\begin{equation}
\require{AMScd}
\begin{CD}
S' @>^{66} >> Y\\
@V^{55}VV @VV V\\
S @> >> X
\end{CD}
\end{equation}
</p>
</body>
</html>

Commutative diagrams II - XyJax (or XyPic.js) environment (very useful)

As

$$ \begin{xy} \xymatrix { U \ar@/_/[ddr]_y \ar@{.>}[dr]|{\langle x,y \rangle} \ar@/^/[drr]^x \\ & X \times_Z Y \ar[d]^q \ar[r]_p & X \ar[d]_f \\ & Y \ar[r]^g & Z } \end{xy} $$

given by

<html>
<head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width">
  <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
  <script id="MathJax-script" async
          src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
  </script>
</head>
<body>
<p>
$$
\begin{xy}
\xymatrix {
U \ar@/_/[ddr]_y \ar@{.>}[dr]|{\langle x,y \rangle} \ar@/^/[drr]^x \\
 & X \times_Z Y \ar[d]^q \ar[r]_p & X \ar[d]_f \\
 & Y \ar[r]^g & Z
}
\end{xy}
$$
</p>
</body>
</html>

Others

Adding tables

Name Upvotes Downvotes
Totals 21 23
Alice 10 11
Bob 4 3
Charlie 7 9

given by

<table>
  <thead>
    <tr>
      <th>Name</th>
      <th>Upvotes</th>
      <th>Downvotes</th>
    </tr>
  </thead>
  <tfoot>
    <tr>
      <td>Totals</td>
      <td>21</td>
      <td>23</td>
    </tr>
  </tfoot>
  <tbody>
    <tr>
      <td>Alice</td>
      <td>10</td>
      <td>11</td>
    </tr>
    <tr>
      <td>Bob</td>
      <td>4</td>
      <td>3</td>
    </tr>
    <tr>
      <td>Charlie</td>
      <td>7</td>
      <td>9</td>
    </tr>
  </tbody>
</table>

Adding pictures

After upload a picture, we get:

Text

given by ![Text](/my_pics/text1.jpg).

Adding footnotes

I’m happy1!

given by

I'm happy[^1]!

[^1]: Well lucky!

Something useful

Markdown codes.

Excerpt

We need add this at the first place of our blog:

---
layout: post
title: Text the excerptions
date: 2022-11-14
last_modified_at: 2022-11-14
tags: [others]
toc:  true
math: true
excerpt_separator: <!--more-->
---

Then we need to write

<!--more-->

to end the excerption.

Comments

See A new comments system for my static Jekyll site.

  1. Well lucky! ↩

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