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<div><h1>Schubert2 -- A package for computations in Intersection Theory</h1>
<div class="single"><h2>Description</h2>
<div><div>The primary purpose of this package is to help compute with intersection theory on smooth varieties. An <a href="___Abstract__Variety.html" title="The Schubert2 data type of an abstract variety">AbstractVariety</a> is not given by equations. Instead, one gives its intersection ring (usually mod numerical equivalence), its dimension, and the chern class of its tangent bundle. An <a href="___Abstract__Sheaf.html" title="the class of sheaves given by their Chern classes">AbstractSheaf</a> is represented by its total chern class (or by its chern character). An <tt>AbstractVarietyMap</tt> (missing documentation<!-- tag: AbstractVarietyMap -->) is a ’morphism’ of abstract varieties, and the information encoded is the pull-back to the corresponding intersection rings.</div>
<div>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>
</div>
<div class="single"><h2>Authors</h2>
<ul><li><div class="single"><a href="http://www.math.uiuc.edu/~dan/">Daniel R. Grayson</a><span> <<a href="mailto:dan@math.uiuc.edu">dan@math.uiuc.edu</a>></span></div>
</li>
<li><div class="single"><a href="http://www.math.cornell.edu/People/Faculty/stillman.html">Michael E. Stillman</a><span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
</li>
<li><div class="single">Stein A. Strømme<span> <<a href="mailto:stromme@math.uib.no">stromme@math.uib.no</a>></span></div>
</li>
<li><div class="single"><a href="http://www.msri.org/~de/">David Eisenbud</a><span> <<a href="mailto:de@msri.org">de@msri.org</a>></span></div>
</li>
</ul>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>0.2</b> of Schubert2.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/Schubert2.m2">Schubert2.m2</a>. The auxiliary files accompanying it are in the directory <a href="../../../../Macaulay2/Schubert2/">Schubert2/</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<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><a href="___Abstract__Variety.html" title="The Schubert2 data type of an abstract variety">AbstractVariety</a> -- The Schubert2 data type of an abstract variety</span></li>
<li><span><tt>AbstractVarietyMap</tt> (missing documentation<!-- tag: AbstractVarietyMap -->)</span></li>
<li><span><tt>ChernClassSymbol</tt> (missing documentation<!-- tag: ChernClassSymbol -->)</span></li>
<li><span><tt>FlagBundle</tt> (missing documentation<!-- tag: FlagBundle -->)</span></li>
<li><span><tt>FlagBundleStructureMap</tt> (missing documentation<!-- tag: FlagBundleStructureMap -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><tt>abstractSheaf</tt> (missing documentation<!-- tag: abstractSheaf -->)</span></li>
<li><span><tt>abstractVariety</tt> (missing documentation<!-- tag: abstractVariety -->)</span></li>
<li><span><tt>adams</tt> (missing documentation<!-- tag: adams -->)</span></li>
<li><span><a href="_base.html" title="an abstract variety, defined with some parameters and some bundles">base</a> -- an abstract variety, defined with some parameters and some bundles</span></li>
<li><span><tt>ch</tt> (missing documentation<!-- tag: ch -->)</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>ctop</tt> (missing documentation<!-- tag: ctop -->)</span></li>
<li><span><tt>expp</tt> (missing documentation<!-- tag: expp -->)</span></li>
<li><span><tt>flagBundle</tt> (missing documentation<!-- tag: flagBundle -->)</span></li>
<li><span><tt>integral</tt> (missing documentation<!-- tag: integral -->)</span></li>
<li><span><tt>intersectionRing</tt> (missing documentation<!-- tag: intersectionRing -->)</span></li>
<li><span><tt>logg</tt> (missing documentation<!-- tag: logg -->)</span></li>
<li><span><tt>lowerstar</tt> (missing documentation<!-- tag: lowerstar -->)</span></li>
<li><span><tt>projectiveBundle</tt> (missing documentation<!-- tag: projectiveBundle -->)</span></li>
<li><span><a href="_projective__Space.html" title="Makes an AbstractVariety representing projective space">projectiveSpace</a> -- Makes an AbstractVariety representing projective space</span></li>
<li><span><tt>reciprocal</tt> (missing documentation<!-- tag: reciprocal -->)</span></li>
<li><span><a href="_schubert__Cycle.html" title="Schubert Cycles on a Grassmannian in terms of Chern classes of the Tautological bundle.">schubertCycle</a> -- Schubert Cycles on a Grassmannian in terms of Chern classes of the Tautological bundle.</span></li>
<li><span><tt>schur</tt> (missing documentation<!-- tag: schur -->)</span></li>
<li><span><tt>sectionClass</tt> (missing documentation<!-- tag: sectionClass -->)</span></li>
<li><span><tt>segre</tt> (missing documentation<!-- tag: segre -->)</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>
</div>
</li>
<li><div class="single">Methods<ul><li><span><tt>AbstractSheaf * AbstractSheaf</tt> (missing documentation<!-- tag: (*,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ** AbstractSheaf</tt> (missing documentation<!-- tag: (**,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf + AbstractSheaf</tt> (missing documentation<!-- tag: (+,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf + ZZ</tt> (missing documentation<!-- tag: (+,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf ++ AbstractSheaf</tt> (missing documentation<!-- tag: (++,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ++ ZZ</tt> (missing documentation<!-- tag: (++,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf - AbstractSheaf</tt> (missing documentation<!-- tag: (-,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ^ ZZ</tt> (missing documentation<!-- tag: (^,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf ^** QQ</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,QQ) -->)</span></li>
<li><span><tt>AbstractSheaf ^** RingElement</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,RingElement) -->)</span></li>
<li><span><tt>AbstractSheaf ^** ZZ</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf QQ</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,QQ) -->)</span></li>
<li><span><tt>AbstractSheaf RingElement</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,RingElement) -->)</span></li>
<li><span><tt>AbstractSheaf ZZ</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>abstractSheaf(AbstractVariety,RingElement)</tt> (missing documentation<!-- tag: (abstractSheaf,AbstractVariety,RingElement) -->)</span></li>
<li><span><tt>adams(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (adams,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>ch(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (ch,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>chern(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (chern,ZZ,AbstractSheaf) -->)</span></li>
<li><span><a href="_chern_lp__Z__Z_cm__Z__Z_cm__Abstract__Sheaf_rp.html" title="Get the Chern class of an abstract sheaf">chern(ZZ,ZZ,AbstractSheaf)</a> -- Get the Chern class of an abstract sheaf</span></li>
<li><span><tt>exteriorPower(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (exteriorPower,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>FlagBundle _ List</tt> (missing documentation<!-- tag: (_,FlagBundle,List) -->)</span></li>
<li><span><tt>FlagBundle _ Sequence</tt> (missing documentation<!-- tag: (_,FlagBundle,Sequence) -->)</span></li>
<li><span><tt>flagBundle(List,AbstractSheaf)</tt> (missing documentation<!-- tag: (flagBundle,List,AbstractSheaf) -->)</span></li>
<li><span><tt>flagBundle(List,AbstractVariety)</tt> (missing documentation<!-- tag: (flagBundle,List,AbstractVariety) -->)</span></li>
<li><span><tt>Hom(AbstractSheaf,AbstractSheaf)</tt> (missing documentation<!-- tag: (Hom,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>lowerstar(AbstractVarietyMap,Thing)</tt> (missing documentation<!-- tag: (lowerstar,AbstractVarietyMap,Thing) -->)</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>schubertCycle(FlagBundle,List)</tt> (missing documentation<!-- tag: (schubertCycle,FlagBundle,List) -->)</span></li>
<li><span><tt>schubertCycle(FlagBundle,Sequence)</tt> (missing documentation<!-- tag: (schubertCycle,FlagBundle,Sequence) -->)</span></li>
<li><span><tt>schur(List,AbstractSheaf)</tt> (missing documentation<!-- tag: (schur,List,AbstractSheaf) -->)</span></li>
<li><span><tt>segre(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (segre,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(QQ,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,QQ,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(RingElement,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,RingElement,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>ZZ + AbstractSheaf</tt> (missing documentation<!-- tag: (+,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>ZZ ++ AbstractSheaf</tt> (missing documentation<!-- tag: (++,ZZ,AbstractSheaf) -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Symbols<ul><li><span><tt>Base</tt> (missing documentation<!-- tag: Base -->)</span></li>
<li><span><tt>Bundle</tt> (missing documentation<!-- tag: Bundle -->)</span></li>
<li><span><tt>BundleRanks</tt> (missing documentation<!-- tag: BundleRanks -->)</span></li>
<li><span><tt>Bundles</tt> (missing documentation<!-- tag: Bundles -->)</span></li>
<li><span><tt>ChernCharacter</tt> (missing documentation<!-- tag: ChernCharacter -->)</span></li>
<li><span><tt>ChernClass</tt> (missing documentation<!-- tag: ChernClass -->)</span></li>
<li><span><tt>IntersectionRing</tt> (missing documentation<!-- tag: IntersectionRing -->)</span></li>
<li><span><tt>PullBack</tt> (missing documentation<!-- tag: PullBack -->)</span></li>
<li><span><tt>PushForward</tt> (missing documentation<!-- tag: PushForward -->)</span></li>
<li><span><tt>QuotientBundles</tt> (missing documentation<!-- tag: QuotientBundles -->)</span></li>
<li><span><tt>Rank</tt> (missing documentation<!-- tag: Rank -->)</span></li>
<li><span><tt>SectionClass</tt> (missing documentation<!-- tag: SectionClass -->)</span></li>
<li><span><tt>StructureMap</tt> (missing documentation<!-- tag: StructureMap -->)</span></li>
<li><span><tt>SubBundles</tt> (missing documentation<!-- tag: SubBundles -->)</span></li>
<li><span><tt>TangentBundle</tt> (missing documentation<!-- tag: TangentBundle -->)</span></li>
<li><span><tt>TautologicalLineBundle</tt> (missing documentation<!-- tag: TautologicalLineBundle -->)</span></li>
<li><span><tt>ToddClass</tt> (missing documentation<!-- tag: ToddClass -->)</span></li>
<li><span><tt>VariableName</tt> (missing documentation<!-- tag: VariableName -->)</span></li>
<li><span><tt>VariableNames</tt> (missing documentation<!-- tag: VariableNames -->)</span></li>
<li><span><tt>VarietyDimension</tt> (missing documentation<!-- tag: VarietyDimension -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Other things<ul><li><span><tt>point</tt> (missing documentation<!-- tag: point -->)</span></li>
<li><span><tt>PP</tt> (missing documentation<!-- tag: PP -->)</span></li>
</ul>
</div>
</li>
</ul>
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