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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>Toric</h1>


<h2>toric varieties and some combinatorial geometry computations</h2>

<p>
    1.9.6</p>

<p>
    4 July 2024
  </p>

</div>
<p><b>
    David Joyner




  </b>
<br />Email: <span class="URL"><a href="mailto:wdjoyner@gmail.com">wdjoyner@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="https://sites.google.com/site/wdjoyner/">https://sites.google.com/site/wdjoyner/</a></span>
<br />Address: <br />W. David Joyner<br /> Mathematics Department<br /> U. S. Naval Academy<br /> Annapolis, MD 21402<br /> USA<br />
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2004-2017 David Joyner.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The code for the <strong class="pkg">toric</strong> package was written during the summer of 2002. It was put into <strong class="pkg">GAP</strong> package format in the summer of 2004.</p>

<p><strong class="pkg">toric</strong> is free software; you can redistribute it and/or modify it under the terms of the MIT License.</p>

<p><strong class="pkg">toric</strong> is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the MIT License for more details.</p>

<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> package of Frank Lübeck and Max Neunhöffer. Moreover, a bug in toric 1.8 was fixed with the help of Max Horn, and this documentation was modified accordingly. Finally, I thank Olexandr Konovalov and Max Horn for transferring this package to the new Git repository.</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A77DB9F7E392A98">1.1 <span class="Heading">Introduction to the <strong class="pkg">toric</strong> package</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7C4637B9828E445B">1.2 <span class="Heading">Introduction to constructing toric varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7AF8D94A7E56C049">1.2-1 <span class="Heading">Generalities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A87B1F97D958BA9">1.2-2 <span class="Heading">Basic combinatorial geometry constructions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X857707BA7D2336A0">1.2-3 <span class="Heading">Basic affine toric variety constructions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86627F4181E72808">1.2-4 <span class="Heading">Riemann-Roch spaces and related constructions</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap2.html#X7D23D3CC7F0A06BA">2 <span class="Heading">Cones and semigroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8524A7567BA4FFA6">2.1 <span class="Heading">Cones</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FEBB7547EEE8E2A">2.1-1 InsideCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87566480802A161C">2.1-2 InDualCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B303CDE8729008F">2.1-3 PolytopeLatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X872AD1E785C7EB03">2.1-4 Faces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A2DA9B38507BDD3">2.1-5 ConesOfFan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C923A4B785606D6">2.1-6 NumberOfConesOfFan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80C858E97E741B21">2.1-7 ToricStar</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80AF5F307DBDC2B4">2.2 <span class="Heading">Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X818998428722C3B5">2.2-1 DualSemigroupGenerators</a></span>
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<div class="ContChap"><a href="chap3.html#X82F418F483E4D0D6">3 <span class="Heading">Affine toric varieties</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B54D98C7A1AC612">3.1 <span class="Heading">Ideals defining affine toric varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8139AACB7F0F44EE">3.1-1 EmbeddingAffineToricVariety</a></span>
</div></div>
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<div class="ContChap"><a href="chap4.html#X807434BE8602C8A5">4 <span class="Heading">Toric varieties <span class="SimpleMath">X(∆)</span> </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7E9ACBE683770EAE">4.1 <span class="Heading">Riemann-Roch spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X802CEF058114DF72">4.1-1 DivisorPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82A512AB7E8F897A">4.1-2 DivisorPolytopeLatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F7ECE28858FE070">4.1-3 RiemannRochBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7EE437E17C7331B7">4.2 <span class="Heading">Topological invariants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8307F8DB85F145AE">4.2-1 EulerCharacteristic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87FB8EBC7FBD8B95">4.2-2 BettiNumberToric</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X80D0D8F07CF1BE07">4.3 <span class="Heading">Points over a finite field</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8289500778E8DE0E">4.3-1 CardinalityOfToricVariety</a></span>
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</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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