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<?xml Add(KK1, 1To[osition(P, P[i])][p+Position(P,P[j])][2][1]+1*To[Position(P, K[i])][p+Position(P,K[j])][2][1]));

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<p>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div class="ChapSects"><a href="chap5.html#X7A489A5D79DA9E5C">5 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X81A7E0D380CE7F31">5.1 <span classK +[java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 113
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.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.html#X7E32106D7B13B8D9">5.1-2 <span class="Heading">DE-java.lang.StringIndexOutOfBoundsException: Range [0, 140) out of bounds for length 36
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.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.html#X7E8F44338461DC08">5.1-4 <span class="Heading">Schenck-8.3</span></a>
<>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.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.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.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.html#X86AF934C83004BF2">5.3 <span java.lang.StringIndexOutOfBoundsException: Range [0, 131) out of bounds for length 4
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.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.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=" if (p+Position(PKj]-K[] =p+(,[]-i 

<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="if (t[Position(P,P[j]-P[i] )] = 1) then
<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">ap&;/span> <pan ="GAPinput"> :=BettiTable(d );/pan>
<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="else
<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 Add(KK2, (+*oPosition(,Pi)[+osition(,[]]2]1-*[PositionP,]p+osition(P,[j]]2][])java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
 total:  3 8 6 1
----------------
     2:  3 8 6 .
     3:  . . . 1
----------------
degree:  0 1 2 3
<pre<divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<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>
<panclass=GAPprompt>gap&t/pan <pan class=GAPinput> : RightPresentationWithDegrees(mat S )</pan>
<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>
 :  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">Add(, (+1*o[osition(P, P[i)][p+(P,[j)[]1+*[(,K[]]pPositionPKj)[2]])
<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.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 
<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    
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><(,+(, []Ki))
 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"Add(KK2, (+1*To[Position(P, P[i])][p+Position(P,P[j])][2][1]-1*To[Position(P, K[i])][p+Position(P,K[j])][2][1]));
<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"gapgt;<span><span class=GAPinput> :=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.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="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 7
<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>
lt;A right   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.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 if(+Position(,[]K[])>pPosition(,[j]P[]) then
<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 := Add(KK1, (+1*To[Position(P, P[i])][Position(P,P[j])][2][1]-1*To[Position(P, K[osition(,K[])]2[1]))java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
<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 classAdd(K1 +Position(,Kj][i ;
<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>
& 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>
< class="APprompt"gap&t<span>< ="APinput>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"></java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 4

<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="Add(, (+1*o[Position(, P[i])][osition(P,P[j])][2][1]+1*To[Position(P, K[i])][Position(P,K[j])][2][1]));
<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="Add(KK1,+java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 36
<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></ if (+P,j+[)=Position,P]Pi)java.lang.StringIndexOutOfBoundsException: Range [60, 61) out of bounds for length 60
<span class="GAPprompt">gap></span> <span class="GAPinput">wmor := GradedMap( wmat, "free", "free", "left", S );;</span>
<span class="GAPprompt">gap></span> <span class="if (t[Position(,[j]P[] ) =1)then
<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>
  yetunknown  6generatorsg;
<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> 
<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 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<A homomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">NaturalMapToModuleOfGlobalSections( W );</span>
A java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 45
</pre></div>

<p><a id="X7DD8F76D7A4206E3" name="X7DD8F76D7A4206E3"></a></p>

<h5>5.3-2 <span class="Heading">Horrocks Mumford bundle</span></h5>

<>This computes theglobalsections module   HorrocksMumford bundle</>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "GradedRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R :fi 
<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>
<panclass=GAPprompt"gap&t;</pan><span class="APinput>( "radedModules");/pan>
<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><java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 69
<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">Add(KK1, -1*(+1*To[Position(P, P[i])][Position(P,P[j])][2][1]-1*T,Ki)[java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 112
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">M := GuessModuleOfGlobalSectionsFromATateMapK ( [) );
#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 Add(K2 1(1ToPositionP,[i])[osition(,P[]][][1]-*o[Position(P, Ki)[(P,K[j)[]1)
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>
 :19 35 20  2
--------------------
     3:   4  .  .  .
     4:  15 35 20  .
     5:   .  .  .  2
--------------------
degree:   0  1  2  3
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 4
total:  100  37  14  10   5   2   5  10  14  37 100   ?   ?   ?   ?
----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
    .   .   .   .     .   .  .  00      
    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">apgt;<span span class"APinput"P:ElementOfGrothendieckGroup )<>
( 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) ) -> java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 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">fi 
( 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!fi java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<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 ]] 1/12t+2/3*^-/2*^2-7/*-5 java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contentsif not (IsEmpty(To[Position(P, P[i])][p+Position(P,P[j])][2])=true) then


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chapBib.html">Bib</a>  <aif(+(K-]  PositionP[][])then

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<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
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