http://stephen.period-domain.com/Recent content on Stephen’s period domainHugoen-ussicheng.liu@temple.edu (Stephen Liu)sicheng.liu@temple.edu (Stephen Liu)Wed, 03 Jun 2026 23:50:29 -0400http://stephen.period-domain.com/posts/segre/Wed, 03 Jun 2026 23:50:29 -0400sicheng.liu@temple.edu (Stephen Liu)http://stephen.period-domain.com/posts/segre/<h2 id="segre-embedding">Segre embedding</h2> <p>Here is the affine part of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>-points of the embedding of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">P</mi><mn>1</mn></msup><mo>×</mo><msup><mi mathvariant="double-struck">P</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^1\times\mathbb{P}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathbb">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span> into <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">P</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>.</p> <div class="sagecell-container"> <script type="text/x-sage"> var('x, y, z') G=implicit_plot3d(x*y==z, (x,-1,1), (y,-1,1), (z,-1,1),smooth=True, plot_points=60) G.show() </script> </div> <div class="sagecell-container"> <script type="text/x-sage"> var('x, y, z') G=implicit_plot3d(x^4+y^4+z^4-x^2-y^2-z^2-x^2*y^2-x^2*z^2-y^2*z^2+1==0 , (x,-2,2), (y,-2,2), (z,-2,2),smooth=True, plot_points=60) G.show() </script> </div> <p>Hmmm.</p>http://stephen.period-domain.com/about/Wed, 03 Jun 2026 22:21:15 -0400sicheng.liu@temple.edu (Stephen Liu)http://stephen.period-domain.com/about/<p>My name is <strong>Stephen</strong> Sicheng <strong>Liu</strong>, 劉思承. I&rsquo;m a third year PhD student in the department of mathematics at Temple University, advised by Matthew Stover. My current interest lies in the intersection between geometric topology and complex algebraic geometry.</p> <h2 id="math">Math</h2> <h3 id="publications">Publications</h3> <p>Tell you about them later.</p> <h3 id="notes">Notes</h3> <div class="article-list"> <p> <span class="date">2024-12-16</span> <span class="sep"> | </span> <a href="http://stephen.period-domain.com/about/Hecke_CM.pdf">Hecke theta functions and modularity of CM elliptic curves</a> <br/> </p> <p> <span class="date">2024-03-02</span> <span class="sep"> | </span> The Galois group of the <a href="http://stephen.period-domain.com/about/Alg_clos.pdf">algebraic closure of finite fields</a> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="double-struck">Z</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\mathbb{Z}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9523em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9523em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbb">Z</span></span><span style="top:-3.2579em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> <br/> </p>