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<head><title>deformationsFace -- Compute the deformations associated to a face.</title>
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<div><h1>deformationsFace -- Compute the deformations associated to a face.</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>deformationsFace(F,C)</tt><br/><tt>deformationsFace(F,C,I)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Face.html">face</a></span></span></li>
<li><span><tt>C</tt>, <span>an <a href="___Complex.html">embedded complex</a></span></span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, reduced monomial</span></li>
</ul>
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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>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p>Compute the homogeneous (i.e., <a href="_degree_lp__First__Order__Deformation_rp.html" title="The small torus degree of a deformation.">degree(FirstOrderDeformation)</a> zero) deformations associated to a face F of the complex C.</p>
<p>The additional parameter I should be the Stanley-Reisner ideal of C and can be given to avoid computation of the Stanley-Reisner ideal if it is already known. Usually this is not necessary: Once I is computed it is stored in C.ideal, so deformationsFace(F,C,I) is equivalent to deformationsFace(F,C). Note also that all methods producing a complex from an ideal (like <a href="_ideal__To__Complex.html" title="The complex associated to a reduced monomial ideal.">idealToComplex</a>) store the ideal in C.ideal.</p>
<p>The deformations and C are stored in F.deform = C, deformations. Note that usually C is not ofComplex F.</p>
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<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I=ideal(x_0*x_1*x_2,x_3*x_4)
o2 = ideal (x x x , x x )
0 1 2 3 4
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : C1=idealToComplex I
o3 = 2: x x x x x x x x x x x x x x x x x x
0 1 3 0 2 3 1 2 3 0 1 4 0 2 4 1 2 4
o3 : complex of dim 2 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 9, 6, 0, 0}, Euler = 1</pre>
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<tr><td><pre>i4 : F=C1.fc_0_0
o4 = x
0
o4 : face with 1 vertex</pre>
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<tr><td><pre>i5 : deformationsFace(F,C1)
2 2
x x x x x x
0 0 0 0 0 0
o5 = {--, --, --, --, ----, ----}
x x x x x x x x
4 3 2 1 3 4 1 2
o5 : List</pre>
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<tr><td><pre>i6 : F=C1.fc_0_1
o6 = x
1
o6 : face with 1 vertex</pre>
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<tr><td><pre>i7 : deformationsFace(F,C1)
2 2
x x x x x x
1 1 1 1 1 1
o7 = {--, --, --, --, ----, ----}
x x x x x x x x
4 3 2 0 3 4 0 2
o7 : List</pre>
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<tr><td><pre>i8 : F=C1.fc_1_0
o8 = x x
0 1
o8 : face with 2 vertices</pre>
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<tr><td><pre>i9 : deformationsFace(F,C1)
x x
0 1
o9 = {----}
x x
3 4
o9 : List</pre>
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<tr><td><pre>i10 : F=C1.fc_2_0
o10 = x x x
0 1 3
o10 : face with 3 vertices</pre>
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<tr><td><pre>i11 : deformationsFace(F,C1)
o11 = {}
o11 : List</pre>
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<p></p>
<div/>
<table class="examples"><tr><td><pre>i12 : R=QQ[x_0..x_4]
o12 = R
o12 : PolynomialRing</pre>
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<tr><td><pre>i13 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)
o13 = ideal (x x , x x , x x , x x , x x )
0 1 1 2 2 3 3 4 0 4
o13 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i14 : C1=idealToComplex I
o14 = 1: x x x x x x x x x x
0 2 0 3 1 3 1 4 2 4
o14 : complex of dim 1 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1</pre>
</td></tr>
<tr><td><pre>i15 : F=C1.fc_0_1
o15 = x
1
o15 : face with 1 vertex</pre>
</td></tr>
<tr><td><pre>i16 : deformationsFace(F,C1)
2
x x x
1 1 1
o16 = {--, --, ----}
x x x x
4 3 3 4
o16 : List</pre>
</td></tr>
<tr><td><pre>i17 : F=C1.fc_1_1
o17 = x x
0 3
o17 : face with 2 vertices</pre>
</td></tr>
<tr><td><pre>i18 : deformationsFace(F,C1)
o18 = {}
o18 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>To homogenize the denominators of deformations (which are supported inside the link) we use globalSections to deal with the toric case. Speed of this should be improved. For ordinary projective space globalSections works much faster.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a> -- Compute the deformations associated to a Stanley-Reisner complex.</span></li>
<li><span><a href="_link.html" title="The link of a face of a complex.">link</a> -- The link of a face of a complex.</span></li>
<li><span><a href="_global__Sections.html" title="The global sections of a toric divisor.">globalSections</a> -- The global sections of a toric divisor.</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>deformationsFace</tt> :</h2>
<ul><li>deformationsFace(Face,Complex)</li>
<li>deformationsFace(Face,Complex,Ideal)</li>
</ul>
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