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<div class="section" id="topintersections-computing-top-intersections-on-mbar-g-n">
<h1>topintersections: Computing top intersections on Mbar_g,n<a class="headerlink" href="#topintersections-computing-top-intersections-on-mbar-g-n" title="Permalink to this headline">¶</a></h1>
<p>This page demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found <a class="reference external" href="https://github.com/uberparagon/mgn/blob/master/topintersections/tex/Mgn.pdf">here</a>. If you have any questions or problems, or if you find the code useful, please contact the author by email: <code class="docutils literal"><span class="pre">werd2.718@gmail.com</span></code>.</p>
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<div class="section" id="acknowledgments">
<h1>Acknowledgments<a class="headerlink" href="#acknowledgments" title="Permalink to this headline">¶</a></h1>
<p>I worked on this project as a graduate student at Brigham Young University while being funded by the NSA grant of my adviser, Tyler Jarvis. Dr. Jarvis also answered many questions and pointed me to helpful resources during this project. The mathematical basis of the algorithms implemented here come from the paper accompanying Carel Faber’s Maple code, and also from Stephanie Yang’s write up about her Macaulay 2 code. The code computes the Witten tau function using the recursion of Liu and Xu.</p>
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<div class="section" id="installing-and-starting">
<h1>Installing and starting<a class="headerlink" href="#installing-and-starting" title="Permalink to this headline">¶</a></h1>
<p><code class="docutils literal"><span class="pre">topintersections</span></code> is now distributed as part of the <code class="docutils literal"><span class="pre">mgn</span></code> package. <a class="reference external" href="https://github.com/uberparagon/mgn">Click here</a> for installation instructions or to see the source code.</p>
<p>After installing, load the code in a sage session. If it is installed, you will get a welcome message.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">topintersections</span> <span class="kn">import</span> <span class="o">*</span>
<span class="go">***************************************************************************</span>
<span class="go">Welcome to the intersection number computer for the moduli space of curves!</span>
<span class="go">Written by Drew Johnson, based on Carel Faber's exposition and Maple code.</span>
<span class="go">The main commands are "intnum" to compute the intersection number, and</span>
<span class="go">"space" to select the space. Type help(intnum) or help(space) for more</span>
<span class="go">information.</span>
<span class="go">Type save_data("filename") to save the results computed this session and</span>
<span class="go">load_data("filname") to retrieve previously saved data. The default filename is</span>
<span class="go">"mgn_top_intersect.dat" if not specified.</span>
<span class="go">Ctrl-D to quit.</span>
<span class="go">***************************************************************************</span>
</pre></div>
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<div class="section" id="basic-operation">
<h1>Basic operation<a class="headerlink" href="#basic-operation" title="Permalink to this headline">¶</a></h1>
<p>You can use the space command to set the space you are working in. The first argument is the genus, and the last is the number of marked points. The space command does three things:</p>
<blockquote>
<div><ol class="arabic simple">
<li>It prints a list of classes and indexes for easy reference.</li>
<li>It sets the defaut space. This will be the space that the code will work in if you don’t specify one in the function calls.</li>
<li>It injects the variables names into your namespace so you can use them to make polynomials.</li>
</ol>
</div></blockquote>
<p>The class Dg1m1 is the class corresponding to the reducible boundary divisor where one component has genus 1 and the marked point 1. The class irr corresponds to the class of the irreducible boundary divisor. The classes psi, ka, ch, and la represent psi, kappa, chern character, and lambda classes respectively.</p>
<p>Now we are ready to compute some things. You can type in a polynomial in the classes given by the space command:</p>
<p>If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above.</p>
<p>You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example:</p>
<p>The following syntax should be very similar to Carel Faber’s MgnLb.txt Maple program:</p>
<p>However, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error:</p>
<p>Instead, do something like:</p>
<p>If the degree is not correct, you will know. (The code only computes top intersections.)</p>
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<div class="section" id="some-more-examples">
<h1>Some more examples<a class="headerlink" href="#some-more-examples" title="Permalink to this headline">¶</a></h1>
<p>Here are some of the intersection numbers from Faber’s paper:</p>
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<div class="section" id="options">
<h1>Options<a class="headerlink" href="#options" title="Permalink to this headline">¶</a></h1>
<p>You can suppress the helpful message that tells you what you are computing using the <code class="docutils literal"><span class="pre">confirm</span></code> keyword argument.</p>
<p>You can have the program return zero if the degree is wrong instead of raising an exception by using the <code class="docutils literal"><span class="pre">check_degree</span></code> keyword argument.</p>
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<div class="section" id="saving-and-loading">
<h1>Saving and loading<a class="headerlink" href="#saving-and-loading" title="Permalink to this headline">¶</a></h1>
<p>So far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example <code class="docutils literal"><span class="pre">la1^9</span></code> above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary.</p>
<p>We can save this data to a file to avoid computing it over again in our next session.</p>
<p>If you don’t specify a file name, the data is saved to the file <code class="docutils literal"><span class="pre">mgn_top_intersect.dat</span></code>.</p>
<p>To load a previously saved data file, use the following command:</p>
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<h3><a href="#">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">topintersections: Computing top intersections on Mbar_g,n</a></li>
<li><a class="reference internal" href="#acknowledgments">Acknowledgments</a></li>
<li><a class="reference internal" href="#installing-and-starting">Installing and starting</a></li>
<li><a class="reference internal" href="#basic-operation">Basic operation</a></li>
<li><a class="reference internal" href="#some-more-examples">Some more examples</a></li>
<li><a class="reference internal" href="#options">Options</a></li>
<li><a class="reference internal" href="#saving-and-loading">Saving and loading</a></li>
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