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<head><title>tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety</title>
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<div><h1>tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety</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>tangentSheaf X</tt></div>
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<li><span>Function: <a href="_tangent__Sheaf.html" title="tangent sheaf of a projective variety">tangentSheaf</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>X</tt>, <span>a <a href="___Projective__Variety.html">projective variety</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Minimize => </tt><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <span>default value true</span>, whether to apply <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> to the result before returning it</span></span></li>
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<div class="single"><h2>Description</h2>
<div>Computes the tangent sheaf of the projective variety <tt>X</tt><p/>
Tangent sheaf of the projective plane:<table class="examples"><tr><td><pre>i1 : P = Proj(QQ[a,b,c])
o1 = P
o1 : ProjectiveVariety</pre>
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<tr><td><pre>i2 : TP = tangentSheaf(P)
o2 = image {-2} | a b 0 |
{-2} | -c 0 b |
{-2} | 0 c a |
3
o2 : coherent sheaf on P, subsheaf of OO (2)
P</pre>
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<tr><td><pre>i3 : HH^0(TP(-1))
3
o3 = QQ
o3 : QQ-module, free</pre>
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<tr><td><pre>i4 : HH^1(TP(-3))
1
o4 = QQ
o4 : QQ-module, free</pre>
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Tangent sheaf of a plane nodal and cuspidal curve:<table class="examples"><tr><td><pre>i5 : Node = Proj(QQ[a,b,c]/ideal(b^2*c-a^2*(a+c)))
o5 = Node
o5 : ProjectiveVariety</pre>
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<tr><td><pre>i6 : Cusp = Proj(QQ[a,b,c]/ideal(b^2*c-a^3))
o6 = Cusp
o6 : ProjectiveVariety</pre>
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<tr><td><pre>i7 : TNode = tangentSheaf(Node)
o7 = image {0} | 0 0 |
{-1} | b a2+ac |
{-1} | 2a 2bc |
1 2
o7 : coherent sheaf on Node, subsheaf of OO ++ OO (1)
Node Node</pre>
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<tr><td><pre>i8 : HH^0(TNode)
1
o8 = QQ
o8 : QQ-module, free</pre>
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<tr><td><pre>i9 : HH^1(TNode)
o9 = 0
o9 : QQ-module</pre>
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<tr><td><pre>i10 : TCusp = tangentSheaf(Cusp)
o10 = image {1} | 0 0 |
{-1} | -2a -2b |
{-2} | 3bc 3a2 |
1 1 1
o10 : coherent sheaf on Cusp, subsheaf of OO (-1) ++ OO (1) ++ OO (2)
Cusp Cusp Cusp</pre>
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<tr><td><pre>i11 : HH^0(TCusp)
2
o11 = QQ
o11 : QQ-module, free</pre>
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<tr><td><pre>i12 : HH^1(TCusp)
o12 = 0
o12 : QQ-module</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_cotangent__Sheaf.html" title="cotangent sheaf of a projective variety">cotangentSheaf</a> -- cotangent sheaf of a projective variety</span></li>
<li><span><a href="_sheaf.html" title="make a coherent sheaf">sheaf</a> -- make a coherent sheaf</span></li>
<li><span><a href="___Projective__Variety.html" title="the class of all projective varieties">ProjectiveVariety</a> -- the class of all projective varieties</span></li>
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