<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (toric) - Chapter 3: Affine toric varieties</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body> <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> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <p><a id="X82F418F483E4D0D6" name="X82F418F483E4D0D6"></a></p> <div class="ChapSects"><a href="chap3.html#X82F418F483E4D0D6">3. <span class="Heading">Affine toric varieties</span></a> <div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B54D98C7A1AC612">3.1 <span class="Heading">Ideals defining affine toric varieties</span></a> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79544F5178127E54">3.1-1 IdealAffineToricVariety</a></span> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8139AACB7F0F44EE">3.1-2 EmbeddingAffineToricVariety</a></span> </div> </div> <h3>3. <span class="Heading">Affine toric varieties</span></h3> <p>This chapter concerns <strong class="pkg">toric</strong> commands which deal with the coordinate rings of affine toric varieties U_sigma.</p> <p><a id="X7B54D98C7A1AC612" name="X7B54D98C7A1AC612"></a></p> <h4>3.1 <span class="Heading">Ideals defining affine toric varieties</span></h4> <p><a id="X79544F5178127E54" name="X79544F5178127E54"></a></p> <h5>3.1-1 IdealAffineToricVariety</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IdealAffineToricVariety</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div> <p><em>Input</em>: <var class="Arg">L</var> is a list generating a cone (as in <code class="code">DualSemigroupGenerators</code>). <br /> <em>Output</em>: the <strong class="pkg">GAP</strong> ideal defining the toric variety associated to the cone generated by the vectors in <var class="Arg">L</var>.</p> <p>This computation is not very efficient and should not be used for ideals with many generators. For example, if you take <var class="Arg">L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]];</var> then <code class="code">IdealAffineToricVariety(L);</code> can exhaust GAP's memory allocation.</p> <table class="example"> <tr><td><pre> gap> J:=IdealAffineToricVariety([[1,0],[3,4]]); [ two-sided ideal in PolynomialRing(..., [ x_1, x_2 ]), (3 generators) ] gap> GeneratorsOfIdeal(J); [ -x_2^2+x_1, -x_2^3+x_1^2, -x_2^4+x_1^3 ] </pre></td></tr></table> <p><a id="X8139AACB7F0F44EE" name="X8139AACB7F0F44EE"></a></p> <h5>3.1-2 EmbeddingAffineToricVariety</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> EmbeddingAffineToricVariety</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div> <p><em>Input</em>: <var class="Arg">L</var> is a list generating a cone (as in <code class="code">DualSemigroupGenerators</code>). <br /> <em>Output</em>: the toroidal embedding of X=Spec(<code class="code">IdealAffineToricVariety(L)</code>) (given as a list of multinomials).</p> <table class="example"> <tr><td><pre> gap> phi:=EmbeddingAffineToricVariety([[1,0],[3,4]]); [ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ] gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; gap> phi:=EmbeddingAffineToricVariety(L); [ x_3, x_2, x_1/x_5, x_1/x_6 ] </pre></td></tr></table> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <div class="chlinkbot"><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> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>