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< div class= "chlinktop">< span class= "chlink1">Goto Chapter: </ span>< a href= "chap0_mj.html">Top< /a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
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<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X7A489A5D79DA9E5C">5 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X81A7E0D380CE7F31">5.1 <span class="Heading">Betti Diagrams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8441906E83F6845D">5.1-1 <span class="Heading">DE-2.2</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7E32106D7B13B8D9">5.1-2 <span class="Heading">DE-Code</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X793A69C4805C6819">5.1-3 <span class="Heading">Schenck-3.2</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7E8F44338461DC08">5.1-4 <span class="Heading">Schenck-8.3</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7B672C498385F92F">5.1-5 <span class="Heading">Schenck-8.3.3</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85CF19B87D1C375F">5.2 <span class="Heading">Commutative Algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7EA4CC697C01E080">5.2-1 <span class="Heading">Saturate</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X86AF934C83004BF2">5.3 <span class="Heading">Global Section Modules of the Induced Sheaves</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87EE931187E2226C">5.3-1 <span class="Heading">Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7DD8F76D7A4206E3">5.3-2 <span class="Heading">Horrocks Mumford bundle</span></a>
</span>
</div></div>
</div>
<h3>5 <span class="Heading">Examples</span></h3>
<p><a id="X81A7E0D380CE7F31" name="X81A7E0D380CE7F31"></a></p>
<h4>5.1 <span class="Heading">Betti Diagrams</span></h4>
<p><a id="X8441906E83F6845D" name="X8441906E83F6845D"></a></p>
<h5>5.1-1 <span class="Heading">DE-2.2</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ x0^2, x1^2, x2^2 ]", 1, 3, S ); </span>
<A 1 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentationWithDegrees( mat, S );</span>
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentationWithDegrees( mat );</span>
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">d := Resolution( M );</span>
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti := BettiTable( d );</span>
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti );</span>
total: 1 3 3 1
----------------
0: 1 . . .
1: . 3 . .
2: . . 3 .
3: . . . 1
----------------
degree: 0 1 2 3
<span class="GAPprompt">gap></span> <span class="GAPinput">## we are still below the Castelnuovo-Mumford regularity, which is 3:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M2 := SubmoduleGeneratedByHomogeneousPart( 2, M );</span>
<A graded torsion right submodule given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">d2 := Resolution( M2 );</span>
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti2 := BettiTable( d2 );</span>
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti2 );</span>
total: 3 8 6 1
----------------
2: 3 8 6 .
3: . . . 1
----------------
degree: 0 1 2 3
</pre></div>
<p><a id="X7E32106D7B13B8D9" name="X7E32106D7B13B8D9"></a></p>
<h5>5.1-2 <span class="Heading">DE-Code</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ x0^2, x1^2 ]", 1, 2, S );</span>
<A 1 x 2 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentationWithDegrees( mat, S );</span>
<A graded cyclic right module on a cyclic generator satisfying 2 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">d := Resolution( M );</span>
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti := BettiTable( d );</span>
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti );</span>
total: 1 2 1
--------------
0: 1 . .
1: . 2 .
2: . . 1
--------------
degree: 0 1 2
<span class="GAPprompt">gap></span> <span class="GAPinput">m := SubmoduleGeneratedByHomogeneousPart( 2, M );</span>
<A graded torsion right submodule given by 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">d2 := Resolution( m );</span>
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti2 := BettiTable( d2 );</span>
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti2 );</span>
2: 4 8 4
--------------
degree: 0 1 2
</pre></div>
<p><a id="X793A69C4805C6819" name="X793A69C4805C6819"></a></p>
<h5>5.1-3 <span class="Heading">Schenck-3.2</span></h5>
<p>This is an example from Section 3.2 in <a href="chapBib_mj.html#biBSch">[Sch03]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mmat := HomalgMatrix( "[ x, x^3 + y^3 + z^3 ]", 1, 2, Qxyz );</span>
<A 1 x 2 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( Qxyz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentationWithDegrees( mmat, S );</span>
<A graded cyclic right module on a cyclic generator satisfying 2 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Mr := Resolution( M );</span>
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">bettiM := BettiTable( Mr );</span>
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( bettiM );</span>
total: 1 2 1
--------------
0: 1 1 .
1: . . .
2: . 1 1
--------------
degree: 0 1 2
<span class="GAPprompt">gap></span> <span class="GAPinput">R := GradedRing( CoefficientsRing( S ) * "x,y,z,w" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nmat := HomalgMatrix( "[ z^2 - y*w, y*z - x*w, y^2 - x*z ]", 1, 3, R );</span>
<A 1 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := RightPresentationWithDegrees( nmat );</span>
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Nr := Resolution( N );</span>
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">bettiN := BettiTable( Nr );</span>
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( bettiN );</span>
total: 1 3 2
--------------
0: 1 . .
1: . 3 2
--------------
degree: 0 1 2
</pre></div>
<p><a id="X7E8F44338461DC08" name="X7E8F44338461DC08"></a></p>
<h5>5.1-4 <span class="Heading">Schenck-8.3</span></h5>
<p>This is an example from Section 8.3 in <a href="chapBib_mj.html#biBSch">[Sch03]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,w";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">jmat := HomalgMatrix( "[ z*w, x*w, y*z, x*y, x^3*z - x*z^3 ]", 1, 5, S );</span>
<A 1 x 5 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">J := RightPresentationWithDegrees( jmat );</span>
<A graded cyclic right module on a cyclic generator satisfying 5 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Jr := Resolution( J );</span>
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti := BettiTable( Jr );</span>
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti );</span>
total: 1 5 6 2
----------------
0: 1 . . .
1: . 4 4 1
2: . . . .
3: . 1 2 1
----------------
degree: 0 1 2 3
</pre></div>
<p><a id="X7B672C498385F92F" name="X7B672C498385F92F"></a></p>
<h5>5.1-5 <span class="Heading">Schenck-8.3.3</span></h5>
<p>This is Exercise 8.3.3 in <a href="chapBib_mj.html#biBSch">[Sch03]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( Qxyz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ x*y*z, x*y^2, x^2*z, x^2*y, x^3 ]", 1, 5, S );</span>
<A 1 x 5 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentationWithDegrees( mat, S );</span>
<A graded cyclic right module on a cyclic generator satisfying 5 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Mr := Resolution( M );</span>
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">betti := BettiTable( Mr );</span>
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( betti );</span>
total: 1 5 6 2
----------------
0: 1 . . .
1: . . . .
2: . 5 6 2
----------------
degree: 0 1 2 3
</pre></div>
<p><a id="X85CF19B87D1C375F" name="X85CF19B87D1C375F"></a></p>
<h4>5.2 <span class="Heading">Commutative Algebra</span></h4>
<p><a id="X7EA4CC697C01E080" name="X7EA4CC697C01E080"></a></p>
<h5>5.2-1 <span class="Heading">Saturate</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := GradedLeftSubmodule( "x,y,z", S );</span>
<A graded torsion-free (left) ideal given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">I := Intersect( m^3, GradedLeftSubmodule( "x", S ) );</span>
<A graded torsion-free (left) ideal given by 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrRelations( I );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">Im := SubobjectQuotient( I, m );</span>
<A graded torsion-free rank 1 (left) ideal given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">I_m := Saturate( I, m );</span>
<A graded principal (left) ideal of rank 1 on a free generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Is := Saturate( I );</span>
<A graded principal (left) ideal of rank 1 on a free generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Assert( 0, Is = I_m );</span>
</pre></div>
<p><a id="X86AF934C83004BF2" name="X86AF934C83004BF2"></a></p>
<h4>5.3 <span class="Heading">Global Section Modules of the Induced Sheaves</span></h4>
<p><a id="X87EE931187E2226C" name="X87EE931187E2226C"></a></p>
<h5>5.3-1 <span class="Heading">Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "GradedRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyzt := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,t";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( Qxyzt );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"></span>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y, y*z, z*t, 0, 0, 0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^3*z,x^2*z^2,0, x*z^2*t, -z^2*t^2, 0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^4, x^3*z, 0, x^2*z*t, -x*z*t^2, 0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x*y, -y^2, x^2-t^2, 0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*z, -x*y*z, y*z*t, 0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*y-x^2*t,-x*y^2+x*y*t,y^2*t-y*t^2,0,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, 0, 0, -1, 1 \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 7, 6, Qxyzt );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"></span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "GradedModules" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmor := GradedMap( wmat, "free", "free", "left", S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( wmor );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentationWithDegrees( wmat, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HW := ModuleOfGlobalSections( W );</span>
<A graded left module presented by yet unknown relations for 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LinearStrandOfTateResolution( W, 0,4 );</span>
<A cocomplex containing 4 morphisms of graded left modules at degrees
[ 0 .. 4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">purity_iso := IsomorphismOfFiltration( PurityFiltration( W ) );</span>
<A non-zero isomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hpurity_iso := ModuleOfGlobalSections( purity_iso );</span>
<An isomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">ModuleOfGlobalSections( wmor );</span>
<A homomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">NaturalMapToModuleOfGlobalSections( W );</span>
<A homomorphism of graded left modules>
</pre></div>
<p><a id="X7DD8F76D7A4206E3" name="X7DD8F76D7A4206E3"></a></p>
<h5>5.3-2 <span class="Heading">Horrocks Mumford bundle</span></h5>
<p>This example computes the global sections module of the Horrocks-Mumford bundle.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "GradedRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x4";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := KoszulDualRing( S, "e0..e4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "GradedModules" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">e1*e4, e2*e0, e3*e1, e4*e2, e0*e3, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">e2*e3, e3*e4, e4*e0, e0*e1, e1*e2 \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]",</span>
<span class="GAPprompt">></span> <span class="GAPinput">2, 5, A );</span>
<A 2 x 5 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := GradedMap( mat, "free", "free", "left", A );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">M := GuessModuleOfGlobalSectionsFromATateMap( 2, phi );</span>
#I GuessModuleOfGlobalSectionsFromATateMap uses a heuristic for efficiency;
please check the correctness of the following result
<A graded left module presented by yet unknown relations for 19 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPure( M );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank( M );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( BettiTable( Resolution( M ) ) );</span>
total: 19 35 20 2
--------------------
3: 4 . . .
4: 15 35 20 .
5: . . . 2
--------------------
degree: 0 1 2 3
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( BettiTable( TateResolution( M, -5, 5 ) ) );</span>
total: 100 37 14 10 5 2 5 10 14 37 100 ? ? ? ?
----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4: 100 35 4 . . . . . . . . 0 0 0 0
3: * . 2 10 10 5 . . . . . . 0 0 0
2: * * . . . . . 2 . . . . . 0 0
1: * * * . . . . . . 5 10 10 2 . 0
0: * * * * . . . . . . . . 4 35 100
----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---S
twist: -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-------------------------------------------------------------------
Euler: 100 35 2 -10 -10 -5 0 2 0 -5 -10 -10 2 35 100
<span class="GAPprompt">gap></span> <span class="GAPinput">M;</span>
<A graded reflexive non-projective rank 2 left module presented by 99 \
relations for 19 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := ElementOfGrothendieckGroup( M );</span>
( 2*O_{P^4} - 1*O_{P^3} - 4*O_{P^2} - 2*O_{P^1} ) -> P^4
<span class="GAPprompt">gap></span> <span class="GAPinput">P!.DisplayTwistedCoefficients := true;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">P;</span>
( 2*O(-3) - 10*O(-2) + 15*O(-1) - 5*O(0) ) -> P^4
<span class="GAPprompt">gap></span> <span class="GAPinput">chi := HilbertPolynomial( M );</span>
1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
<span class="GAPprompt">gap></span> <span class="GAPinput">c := ChernPolynomial( M );</span>
( 2 | 1-h+4*h^2 ) -> P^4
<span class="GAPprompt">gap></span> <span class="GAPinput">ChernPolynomial( M * S^3 );</span>
( 2 | 1+5*h+10*h^2 ) -> P^4
<span class="GAPprompt">gap></span> <span class="GAPinput">ch := ChernCharacter( M );</span>
[ 2-u-7*u^2/2!+11*u^3/3!+17*u^4/4! ] -> P^4
<span class="GAPprompt">gap></span> <span class="GAPinput">HilbertPolynomial( ch );</span>
1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
<span class="GAPprompt">gap></span> <span class="GAPinput">List( [ -8 .. 7 ], i -> Value( chi, i ) );</span>
[ 35, 2, -10, -10, -5, 0, 2, 0, -5, -10, -10, 2, 35, 100, 210, 380 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HF := HilbertFunction( M );</span>
function( t ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">List( [ 0 .. 7 ], HF );</span>
[ 0, 0, 0, 4, 35, 100, 210, 380 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexOfRegularity( M );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">DataOfHilbertFunction( M );</span>
[ [ [ 4 ], [ 3 ] ], 1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5 ]
</pre></div>
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