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### 3 Examples

#### 3.1 Divisionally free but not inductively free arrangement

This is the smallest representable rank $$3$$ matroid which is divisionally free but not inductively free. It has $$14$$ atoms (cf. [BBJ+21]).

gap> #db := AttachMatroidsDatabase( ); # needs arangosh
gap> #key := "ef53049d834fba1b21f36c365d7f1d46d7fce2f2";
gap> #d := db.matroids_split_public.document( key );
gap> d := [ [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ],
>        [ 1, 10, 11, 12 ], [ 2, 6, 10, 13 ], [ 2, 7, 11, 14 ],
>        [ 3, 6, 12, 14 ], [ 3, 8, 11, 13 ], [ 4, 9, 10, 14 ],
>        [ 4, 7, 13 ], [ 5, 7, 12 ], [ 5, 8, 10 ],
>        [ 5, 9, 11 ], [ 5, 13, 14 ], [ 9, 12, 13 ],
>        [ 1, 13 ], [ 1, 14 ], [ 2, 8 ], [ 2, 9 ], [ 2, 12 ], [ 3, 7 ],
>        [ 3, 9 ], [ 3, 10 ], [ 4, 6 ], [ 4, 8 ], [ 4, 11 ], [ 4, 12 ],
>        [ 5, 6 ], [ 6, 11 ], [ 7, 10 ], [ 8, 12 ], [ 8, 14 ] ];;
gap> matroid := MatroidByCoatomsNC( 14, 3, d );
<A matroid>
gap> #homalgIOMode( "f" );
gap> zz := HomalgRingOfIntegersInSingular( );
Z
gap> SetInfoLevel( InfoWarning, 0 );
gap> M := EquationsAndInequationsOfModuliSpaceOfMatroid( matroid, zz );;
gap> LoadPackage( "ZariskiFrames", ">= 2023.06-06" );
true
gap> m := DistinguishedQuasiAffineSet( M );
V_{Z[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]}( I )
\ V_{Z[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]}( J_1 )
\ .. \ V_{Z[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]}( J_199 )
gap> e := EmbedInSmallerAmbientSpace( m );
V_{Z[a9]}( I ) \ V_{Z[a9]}( J_1 ) \ V_{Z[a9]}( J_2 )
gap> Display( e );
V( <2*a9^2-2*a9+1> ) \ V( <5,a9+1> ) \ V( <5,a9-2> )
gap> a := DistinguishedLocallyClosedPart( e );
V_{Z[a9]}( I ) \ V_{Z[a9]}( J_1 ) \ V_{Z[a9]}( J_2 )
gap> Display( a );
V( <2*a9^2-2*a9+1> ) \ V( <3*a9-1> ) \ V( <a9+1> )
gap> piter := PseudoIteratorOfClosedPoints( e );
<iterator of closed points of
V_{Z[a9]}( I ) \ V_{Z[a9]}( J_1 ) \ V_{Z[a9]}( J_2 )>
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(3)[a9]/( a9^2-a9-1 ):
1,0,1,1,    1,  0,1,1,   1,    0, 1,   1, 0,    1,
0,1,1,-a9-1,-a9,0,0,0,   0,    1, a9-1,1, 1,    1,
0,0,0,0,    0,  1,1,a9+1,-a9+1,a9,1,   a9,-a9-1,1

modulo [ a9^2-a9-1 ]
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(7)[a9]/( a9^2-a9-3 ):
1,0,1,1,     1,   0,1,1,      1,    0, 1,      1, 0,     1,
0,1,1,2*a9-1,2*a9,0,0,0,      0,    1, -2*a9+2,1, 1,     1,
0,0,0,0,     0,   1,1,-2*a9+1,-a9+1,a9,1,      a9,2*a9-1,1

modulo [ a9^2-a9-3 ]
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(11)[a9]/( a9^2-a9-5 ):
1,0,1,1,     1,   0,1,1,      1,    0, 1,      1, 0,     1,
0,1,1,2*a9-1,2*a9,0,0,0,      0,    1, -2*a9+2,1, 1,     1,
0,0,0,0,     0,   1,1,-2*a9+1,-a9+1,a9,1,      a9,2*a9-1,1

modulo [ a9^2-a9-5 ]
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(13):
1,0,1,1, 1, 0,1,1,1,0, 1,1, 0, 1,
0,1,1,-5,-4,0,0,0,0,1, 6,1, 1, 1,
0,0,0,0, 0, 1,1,5,3,-2,1,-2,-5,1
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(17):
1,0,1,1,1,0,1,1, 1,0, 1, 1, 0,1,
0,1,1,4,5,0,0,0, 0,1, -3,1, 1,1,
0,0,0,0,0,1,1,-4,7,-6,1, -6,4,1
gap> p := NextIterator( piter );
<A vector matroid>
gap> Display( p );
The vector matroid of this matrix over GF(19)[a9]/( a9^2-a9-9 ):
1,0,1,1,     1,   0,1,1,      1,    0, 1,      1, 0,     1,
0,1,1,2*a9-1,2*a9,0,0,0,      0,    1, -2*a9+2,1, 1,     1,
0,0,0,0,     0,   1,1,-2*a9+1,-a9+1,a9,1,      a9,2*a9-1,1

modulo [ a9^2-a9-9 ]
gap> charset := ConstructibleProjection( e );
( V_{Z}( I1 ) \ V_{Z}( J1_1 ) \ V_{Z}( J1_2 ) )
gap> Display( charset );
( V( <> )

\ V( <2> )

\ V( <5> ) )

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