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<head><title>globalSections -- The global sections of a toric divisor.</title>
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<div><h1>globalSections -- The global sections of a toric divisor.</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>globalSections(A,v)</tt><br/><tt>globalSections(A,v,L)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
<li><span><tt>v</tt>, <span>a <a href="../../Macaulay2Doc/html/___Vector.html">vector</a></span></span></li>
<li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>Computes the global sections of a toric Weil divisor D with coefficients v with respect to the coker grading by A. In the same way as v they are represented by vectors (exponent vectors of Laurent monomials in rank target A variables).</p>
<p>If a list of indices L in 0..rank target A -1 is specified, then those Laurent monomial exponents are computed, which induce a linear equivalence of D to an effective divisor with support precisely on L.</p>
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<table class="examples"><tr><td><pre>i1 : A=matrix {{1, 0}, {0, 1}, {-1, -1}}
o1 = | 1 0 |
| 0 1 |
| -1 -1 |
3 2
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : b=vector {2,0,0}
o2 = | 2 |
| 0 |
| 0 |
3
o2 : ZZ</pre>
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<tr><td><pre>i3 : globalSections(A,b)
o3 = {| -2 |, | -2 |, | -2 |, | -1 |, | -1 |, 0}
| 0 | | 1 | | 2 | | 0 | | 1 |
| 2 | | 1 | | 0 | | 1 | | 0 |
o3 : List</pre>
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<tr><td><pre>i4 : A=matrix {{1, 0}, {0, 1}, {-1, -1},{1,1}}
o4 = | 1 0 |
| 0 1 |
| -1 -1 |
| 1 1 |
4 2
o4 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i5 : b=vector {2,0,0,0}
o5 = | 2 |
| 0 |
| 0 |
| 0 |
4
o5 : ZZ</pre>
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<tr><td><pre>i6 : globalSections(A,b)
o6 = {| -2 |, | -1 |, 0}
| 2 | | 1 |
| 0 | | 0 |
| 0 | | 0 |
o6 : List</pre>
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<tr><td><pre>i7 : globalSections(A,b,{1})
o7 = {| -2 |}
| 2 |
| 0 |
| 0 |
o7 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div><p>This uses the package Polyhedra.m2 (if ConvexInterface.m2 is not present) to compute the lattice points of a convex hull. constructHilbertBasis of the package Polyhedra.m2 used by latticePoints overwrites global variable C. Fixed this in my local version.</p>
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<div class="waystouse"><h2>Ways to use <tt>globalSections</tt> :</h2>
<ul><li>globalSections(Matrix,Vector)</li>
<li>globalSections(Matrix,Vector,List)</li>
</ul>
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