<?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 4: Toric varieties X(Delta) </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="chap3.html">Previous Chapter</a> <a href="chapBib.html">Next Chapter</a> </div>
<p><a id="X7AD3B91A84FFF441" name="X7AD3B91A84FFF441"></a></p>
<div class="ChapSects"><a href="chap4.html#X7AD3B91A84FFF441">4. <span class="Heading">Toric varieties X(Delta) </span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7E9ACBE683770EAE">4.1 <span class="Heading">Riemann-Roch spaces</span></a>
<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 class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7EE437E17C7331B7">4.2 <span class="Heading">Topological invariants</span></a>
<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 class="ContSect"><span class="nocss"> </span><a href="chap4.html#X80D0D8F07CF1BE07">4.3 <span class="Heading">Points over a finite field</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8289500778E8DE0E">4.3-1 CardinalityOfToricVariety</a></span>
</div>
</div>
<h3>4. <span class="Heading">Toric varieties X(Delta) </span></h3>
<p>This chapter concerns <strong class="pkg">toric</strong> commands which deal with certain objects associated to the (non-affine) toric varieties X(Delta).</p>
<p><a id="X7E9ACBE683770EAE" name="X7E9ACBE683770EAE"></a></p>
<h4>4.1 <span class="Heading">Riemann-Roch spaces</span></h4>
<p>Let Delta denote a complete nonsingular fan.</p>
<p><a id="X802CEF058114DF72" name="X802CEF058114DF72"></a></p>
<h5>4.1-1 DivisorPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DivisorPolytope</code>( <var class="Arg">D, Rays</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Rays</var> is the list of smallest integer vectors in the rays for the fan Delta which determine the Weil divisors of X(Delta). <br /> <var class="Arg">D</var> is the list of coefficients for the a Weil divisor. <br /> <em>Output</em>: the linear expressions in the affine coordinates of the space of the cone which must be positive for a point to be in the desired polytope.</p>
<table class="example">
<tr><td><pre>
gap> DivisorPolytope([6,6,0],[[2,-1],[-1,2],[-1,-1]]);
[ 2*x_1-x_2+6, -x_1+2*x_2+6, -x_1-x_2 ]
</pre></td></tr></table>
<p>See also Example 6.13 in <a href="chapBib.html#biBJV02">[JV02]</a>.</p>
<p><a id="X82A512AB7E8F897A" name="X82A512AB7E8F897A"></a></p>
<h5>4.1-2 DivisorPolytopeLatticePoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DivisorPolytopeLatticePoints</code>( <var class="Arg">D, Delta, Rays</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Delta</var> is the fan <br /> <var class="Arg">Rays</var> is the <em>ordered</em> list of rays for <var class="Arg">Delta</var> <br /> <var class="Arg">D</var> is the list of coefficients for a Weil divisor. <br /> <em>Output</em>: the list of points in P_D cap L_0^* which parameterize the elements in the Riemann-Roch space L(D), where P_D is the polytope associated to the divisor D (see <code class="code">DivisorPolytope</code>).</p>
<table class="example">
<tr><td><pre>
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
gap> Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
[ [ -6, -6 ], [ -5, -5 ], [ -5, -4 ], [ -4, -5 ], [ -4, -4 ], [ -4, -3 ],
[ -4, -2 ], [ -3, -4 ], [ -3, -3 ], [ -3, -2 ], [ -3, -1 ], [ -3, 0 ],
[ -2, -4 ], [ -2, -3 ], [ -2, -2 ], [ -2, -1 ], [ -2, 0 ], [ -2, 1 ],
[ -2, 2 ], [ -1, -3 ], [ -1, -2 ], [ -1, -1 ], [ -1, 0 ], [ -1, 1 ],
[ 0, -3 ], [ 0, -2 ], [ 0, -1 ], [ 0, 0 ], [ 1, -2 ], [ 1, -1 ], [ 2, -2 ] ]
gap>
</pre></td></tr></table>
<p><a id="X7F7ECE28858FE070" name="X7F7ECE28858FE070"></a></p>
<h5>4.1-3 RiemannRochBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RiemannRochBasis</code>( <var class="Arg">D, Delta, Rays</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Delta</var> is a complete and nonsingular fan <br /> <var class="Arg">D</var> is the list of coefficients for the Weil divisor<br /> <var class="Arg">Rays</var> is a list of rays for the fan used to describe the Weil divisors. <br /> <em>Output</em>: A basis (a list of monomials) for the Riemann-Roch space of the divisor represented by <var class="Arg">D</var>.</p>
<p>For details on how the Weil divisors can be expressed in terms of the rays of the fan, please see section 3.3 in <a href="chapBib.html#biBF93">[Ful93]</a>. This procedure does not check if the fan is complete and nonsingular.</p>
<table class="example">
<tr><td><pre>
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
gap> Delta:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> RiemannRochBasis(Div,Delta,Rays);
[ 1/(x_1^6*x_2^6), 1/(x_1^5*x_2^5), 1/(x_1^5*x_2^4), 1/(x_1^4*x_2^5),
1/(x_1^4*x_2^4), 1/(x_1^4*x_2^3), 1/(x_1^4*x_2^2), 1/(x_1^3*x_2^4),
1/(x_1^3*x_2^3), 1/(x_1^3*x_2^2), 1/(x_1^3*x_2), 1/x_1^3, 1/(x_1^2*x_2^4),
1/(x_1^2*x_2^3), 1/(x_1^2*x_2^2), 1/(x_1^2*x_2), 1/x_1^2, x_2/x_1^2,
x_2^2/x_1^2, 1/(x_1*x_2^3), 1/(x_1*x_2^2), 1/(x_1*x_2), 1/x_1, x_2/x_1,
1/x_2^3, 1/x_2^2, 1/x_2, 1, x_1/x_2^2, x_1/x_2, x_1^2/x_2^2 ]
</pre></td></tr></table>
<p><a id="X7EE437E17C7331B7" name="X7EE437E17C7331B7"></a></p>
<h4>4.2 <span class="Heading">Topological invariants</span></h4>
<p>Throughout this section, X(Delta) <em>must be non-singular</em>.</p>
<p><a id="X8307F8DB85F145AE" name="X8307F8DB85F145AE"></a></p>
<h5>4.2-1 EulerCharacteristic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> EulerCharacteristic</code>( <var class="Arg">Delta</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Delta</var> is a nonsingular fan of cones, represented by its list of maximal cones. <br /> <em>Output</em>: the Euler characteristic of the toric variety X(Delta), where Delta is a fan determined by <var class="Arg">Delta</var>.</p>
<table class="example">
<tr><td><pre>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> EulerCharacteristic(Cones);
3
</pre></td></tr></table>
<p>Note: X(Delta) <em>must be non-singular</em> here.</p>
<p><a id="X87FB8EBC7FBD8B95" name="X87FB8EBC7FBD8B95"></a></p>
<h5>4.2-2 BettiNumberToric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BettiNumberToric</code>( <var class="Arg">Delta, k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Delta</var> represents a nonsingular fan Delta (represented by maximal cones), <br /> <var class="Arg">k</var> is an integer. <br /> <em>Output</em>: the <var class="Arg">k</var>-th Betti number of the toric variety X(Delta).</p>
<p>The <code class="code">BettiNumberToric</code> procedure does not check if <var class="Arg">Delta</var> is nonsingular. It is possible that this procedure outputs nonsense when <var class="Arg">Delta</var> is not represented by maximal cones or is nonsingular.</p>
<table class="example">
<tr><td><pre>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> BettiNumberToric(Cones,1);
0
gap> BettiNumberToric(Cones,2);
1
gap> Cones:=[[[2,-1],[-1,1]],[[-1,1],[-1,0]],[[-1,0],[2,-1]]];;
gap> BettiNumberToric(Cones,1);
0
gap> BettiNumberToric(Cones,2);
1
</pre></td></tr></table>
<p>Not to be confused with the Betti number of a polycyclically presented torsion free group, already available in <strong class="pkg">GAP</strong>.</p>
<p><a id="X80D0D8F07CF1BE07" name="X80D0D8F07CF1BE07"></a></p>
<h4>4.3 <span class="Heading">Points over a finite field</span></h4>
<p><a id="X8289500778E8DE0E" name="X8289500778E8DE0E"></a></p>
<h5>4.3-1 CardinalityOfToricVariety</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CardinalityOfToricVariety</code>( <var class="Arg">Cones, q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>Input</em>: <var class="Arg">Cones</var> is the list of maximal cones of a fan Delta, <var class="Arg">q</var> is a prime power. <br /> <em>Output</em>: The size of the set of GF(q)-rational points of the toric variety X(Delta).</p>
<p>Note: X(Delta) <em>must be non-singular</em> here.</p>
<table class="example">
<tr><td><pre>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> CardinalityOfToricVariety(Cones,3);
13
gap> CardinalityOfToricVariety(Cones,4);
21
gap> CardinalityOfToricVariety(Cones,5);
31
gap> CardinalityOfToricVariety(Cones,7);
57
</pre></td></tr></table>
<div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap3.html">Previous Chapter</a> <a href="chapBib.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>
distrib
>
Mandriva
>
2010.0
>
i586
>
media
>
contrib-release
>
by-pkgid
>
91213ddcfbe7f54821d42c2d9e091326
>
files
>
2846
