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<head><title>projectiveSpace -- Makes an AbstractVariety representing projective space</title>
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<div><h1>projectiveSpace -- Makes an AbstractVariety representing projective space</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>P=projectiveSpace(n) or P=projectiveSpace(n, baseVariety) or P=projectiveSpace(n, baseVariety, VariableName => h)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>n</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
<li><span><tt>baseVariety</tt>, <span>an <a href="___Abstract__Variety.html">abstract variety</a></span></span></li>
<li><span><tt>h</tt>, <span>a <a href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>P</tt>, <span>an <a href="___Abstract__Variety.html">abstract variety</a></span></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>VariableName => ...</tt> (missing documentation<!-- tag: projectiveSpace(..., VariableName => ...) -->), </span></li>
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<div class="single"><h2>Description</h2>
<div><div>Constructs the projective space <tt>P</tt> of 1-quotients of the trivial bundle on the base variety <tt>baseVariety</tt>. The Chow ring is set to be the polynomial ring over the Chow ring of <tt>baseVariety</tt>, with variable <tt>h</tt>. The tangent bundle of X is available as an <a href="___Abstract__Sheaf.html" title="the class of sheaves given by their Chern classes">AbstractSheaf</a>, accessed by <tt>X.TangentBundle</tt>. Here baseVariety and VariableName are optional.</div>
<table class="examples"><tr><td><pre>i1 : P=projectiveSpace(3)
o1 = P
o1 : a flag bundle with ranks {3, 1}</pre>
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<tr><td><pre>i2 : todd P
11 2 3
o2 = 1 + 2h + --h + h
6
QQ[][H , H , H , h]
1,1 1,2 1,3
o2 : ---------------------------------------------
(H + h, H + H h, H + H h, H h)
1,1 1,2 1,1 1,3 1,2 1,3</pre>
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<tr><td><pre>i3 : chi(OO_P(3))
o3 = 20
o3 : QQ[]</pre>
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<div>If we want a projective space where we can compute the Hilbert Polynomial of a sheaf, we need a variable to represent an integer. We define a base variety that is a point <tt>pt</tt> containing this variable.</div>
<table class="examples"><tr><td><pre>i4 : pt = base(n)
o4 = pt
o4 : an abstract variety of dimension 0</pre>
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<tr><td><pre>i5 : Q=projectiveSpace(4,pt, VariableName => h)
o5 = Q
o5 : a flag bundle with ranks {4, 1}</pre>
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<tr><td><pre>i6 : chi(OO_Q(n))
1 4 5 3 35 2 25
o6 = --n + --n + --n + --n + 1
24 12 24 12
o6 : QQ[n]</pre>
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<div>If be build a projective space over another variety, the dimensions add:</div>
<table class="examples"><tr><td><pre>i7 : baseVariety = projectiveSpace(4, VariableName => h)
o7 = baseVariety
o7 : a flag bundle with ranks {4, 1}</pre>
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<tr><td><pre>i8 : P = projectiveSpace (3,baseVariety, VariableName => H)
o8 = P
o8 : a flag bundle with ranks {3, 1}</pre>
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<tr><td><pre>i9 : dim P
o9 = 7</pre>
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<tr><td><pre>i10 : todd P
5 11 2 35 2 3 55 2 35 2 25 3
o10 = 1 + (2H + -h) + (--H + 5h*H + --h ) + (H + --h*H + --h H + --h ) +
2 6 12 12 6 12
-----------------------------------------------------------------------
5 3 385 2 2 25 3 4 35 2 3 275 3 2 4 25 3 3
(-h*H + ---h H + --h H + h ) + (--h H + ---h H + 2h H) + (--h H +
2 72 6 12 72 12
-----------------------------------------------------------------------
11 4 2 4 3
--h H ) + h H
6
QQ[][H , H , H , H , h]
1,1 1,2 1,3 1,4
-----------------------------------------------------------[H , H , H , H]
(H + h, H + H h, H + H h, H + H h, H h) 1,1 1,2 1,3
1,1 1,2 1,1 1,3 1,2 1,4 1,3 1,4
o10 : --------------------------------------------------------------------------------
(H + H, H + H H, H + H H, H H)
1,1 1,2 1,1 1,3 1,2 1,3</pre>
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<div class="waystouse"><h2>Ways to use <tt>projectiveSpace</tt> :</h2>
<ul><li><span><tt>projectiveSpace(ZZ)</tt> (missing documentation<!-- tag: (projectiveSpace,ZZ) -->)</span></li>
<li><span><tt>projectiveSpace(ZZ,AbstractVariety)</tt> (missing documentation<!-- tag: (projectiveSpace,ZZ,AbstractVariety) -->)</span></li>
</ul>
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