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<div><h1>AbstractVariety -- The Schubert2 data type of an abstract variety</h1>
<div class="single"><h2>Description</h2>
<div><p>An Abstract Variety in Schubert 2 is defined by its dimension and a QQ-algebra, interpreted as the rational Chow ring. For example, the following code defines the abstract variety corresponding to P2, with its Chow ring A. Once the variety X is created, we can access its structure sheaf OO<sub>X</sub>, represented by its Chern class</p>
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<table class="examples"><tr><td><pre>i1 : A=QQ[t]/ideal(t^3)
o1 = A
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : X=abstractVariety(2,A)
o2 = X
o2 : an abstract variety of dimension 2</pre>
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<tr><td><pre>i3 : OO_X
o3 = a sheaf
o3 : an abstract sheaf of rank 1 on X</pre>
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<tr><td><pre>i4 : chern OO_X
o4 = 1
o4 : A</pre>
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<div>A variable of type AbstractVariety is actually of type MutableHashTable, and can contain other information, such as its <tt>TangentBundle</tt> (missing documentation<!-- tag: TangentBundle -->). Once this is defined, we can compute the Todd class.</div>
<table class="examples"><tr><td><pre>i5 : X.TangentBundle = abstractSheaf(X,Rank=>2, ChernClass=>(1+t)^3)
o5 = a sheaf
o5 : an abstract sheaf of rank 2 on X</pre>
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<tr><td><pre>i6 : todd X
3 2
o6 = 1 + -t + t
2
o6 : A</pre>
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<div>If we want things like the Euler characteristic of a sheaf, we must also specify a method to take the <tt>integral</tt> (missing documentation<!-- tag: integral -->) for the Chow ring A; in the case where A is Gorenstein, as is the Chow ring of a complete nonsingular variety, this is a functional that takes the highest degree component. In the following example, The sheaf OO<sub>X</sub> is the structure sheaf of X, and OO<sub>X</sub>(2t) is the line bundle with first Chern class 2t. The computation of the Euler Characteristic is made using the Todd class and the Riemann-Roch formula.</div>
<table class="examples"><tr><td><pre>i7 : integral A := f -> coefficient(t^2,f)
o7 = {*Function[stdio:7:16-7:35]*}
o7 : FunctionClosure</pre>
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<tr><td><pre>i8 : chi(OO_X(2*t))
o8 = 6
o8 : QQ</pre>
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<div>There are several other methods for constructing abstract varieties: the following functions construct basic useful varieties (often returning the corresponding structure map). <a href="_projective__Space.html" title="Makes an AbstractVariety representing projective space">projectiveSpace</a>, <tt>projectiveBundle</tt> (missing documentation<!-- tag: projectiveBundle -->), <tt>flagBundle</tt> (missing documentation<!-- tag: flagBundle -->), <a href="_base.html" title="an abstract variety, defined with some parameters and some bundles">base</a>. Text This package and its documentation are still rather incomplete, but see the examples <a href="___Lines_spon_sphypersurfaces.html" title="Example using Schubert2">Lines on hypersurfaces</a> and <a href="___Conics_spon_spa_spquintic_spthreefold.html" title="Example using Schubert2">Conics on a quintic threefold</a>, which should be enough to figure out some of what’s possible.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Abstract__Sheaf.html" title="the class of sheaves given by their Chern classes">AbstractSheaf</a> -- the class of sheaves given by their Chern classes</span></li>
<li><span><tt>chern</tt> (missing documentation<!-- tag: chern -->)</span></li>
<li><span><tt>chi</tt> (missing documentation<!-- tag: chi -->)</span></li>
<li><span><tt>TangentBundle</tt> (missing documentation<!-- tag: TangentBundle -->)</span></li>
<li><span><tt>todd</tt> (missing documentation<!-- tag: todd -->)</span></li>
</ul>
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<div class="waystouse"><h2>Types of abstract variety :</h2>
<ul><li><span><tt>FlagBundle</tt> (missing documentation<!-- tag: FlagBundle -->)</span></li>
</ul>
<h2>Methods that use an abstract variety :</h2>
<ul><li><span><tt>abstractSheaf(AbstractVariety)</tt> (missing documentation<!-- tag: (abstractSheaf,AbstractVariety) -->)</span></li>
<li><span><tt>abstractSheaf(AbstractVariety,RingElement)</tt> (missing documentation<!-- tag: (abstractSheaf,AbstractVariety,RingElement) -->)</span></li>
<li><span><tt>dim(AbstractVariety)</tt> (missing documentation<!-- tag: (dim,AbstractVariety) -->)</span></li>
<li><span><tt>flagBundle(List,AbstractVariety)</tt> (missing documentation<!-- tag: (flagBundle,List,AbstractVariety) -->)</span></li>
<li><span><tt>intersectionRing(AbstractVariety)</tt> (missing documentation<!-- tag: (intersectionRing,AbstractVariety) -->)</span></li>
<li><span><tt>net(AbstractVariety)</tt> (missing documentation<!-- tag: (net,AbstractVariety) -->)</span></li>
<li><span><tt>OO _ AbstractVariety</tt> (missing documentation<!-- tag: (_,OO,AbstractVariety) -->)</span></li>
<li><span><tt>projectiveBundle(ZZ,AbstractVariety)</tt> (missing documentation<!-- tag: (projectiveBundle,ZZ,AbstractVariety) -->)</span></li>
<li><span><tt>projectiveSpace(ZZ,AbstractVariety)</tt> (missing documentation<!-- tag: (projectiveSpace,ZZ,AbstractVariety) -->)</span></li>
<li><span><tt>tangentBundle(AbstractVariety)</tt> (missing documentation<!-- tag: (tangentBundle,AbstractVariety) -->)</span></li>
<li><span><tt>todd(AbstractVariety)</tt> (missing documentation<!-- tag: (todd,AbstractVariety) -->)</span></li>
<li><span><tt>use(AbstractVariety)</tt> (missing documentation<!-- tag: (use,AbstractVariety) -->)</span></li>
</ul>
<h2>Fixed objects of class AbstractVariety :</h2>
<ul><li><span><tt>point</tt> (missing documentation<!-- tag: point -->)</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Abstract__Variety.html" title="The Schubert2 data type of an abstract variety">AbstractVariety</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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