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<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>
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<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>
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<h4>3.1 <span class="Heading">Ideals defining affine toric varieties</span></h4>
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<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>
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<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>
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