What is the smallest number $n$ of atoms of a representable rank $3$ matroid which is divisionally free but not inductively free?¶
The explanation of the question and the answer $n = 14$ can be found in [BBJKL]. It turns out that there is exactly one representable matroid with $14$ atoms having this property. Below we compute its moduli space of representations and prove that its fibers over $\operatorname{Spec} \mathbb{Z}$ are inhabited in characteristics different from $2$ and $5$.
[BBJKL] Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner. On the generation of rank $3$ simple matroids with an application to Terao's freeness conjecture. (arXiv:1907.01073)
We start by loading the necessary packages:¶
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using CapAndHomalg
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LoadPackage( "MatroidGeneration" )
Construct the matroid out of the list of coatoms, i.e., the flats of rank $n-1 = 3-1 = 2$, see [BBJKL, Example 1.3].¶
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coatoms =
[ [ 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 ] ];
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matroid = MatroidByCoatomsNC( 14, 3, ConvertJuliaToGAP( coatoms ) )
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Define the ring of coefficients over which to define the moduli space of representations:¶
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ℤ = HomalgRingOfIntegersInSingular( )
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SetInfoLevel( InfoWarning, 0 )
Compute the moduli space of representations:¶
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LoadPackage( "ZariskiFrames" )
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M = EquationsAndInequationsOfModuliSpaceOfMatroid( matroid, ℤ );
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m = DistinguishedQuasiAffineSet( M )
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Display( m )
The points of the above quasi-affine moduli space are admissible specializations of the following parametrized arrangement:¶
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arr = ParametrizedObject( m )
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Display( arr )
Embed the moduli space in a significantly smaller ambient affine space:¶
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e = EmbedInSmallerAmbientSpace( m )
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Display( e )
Show that the moduli space is in fact a distinguished quasi affine set, i.e., the difference of an affine set and a hypersurface:¶
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a = DistinguishedLocallyClosedPart( e )
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Display( a )
Rewrite the parametrized arrangement on the parameters of the new ambient affine space:¶
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arr = ParametrizedObject( e )
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Display( arr )
Compute the characteristic set of the matroid, i.e., the projection of the moduli space on $\operatorname{Spec} \mathbb{Z}$:¶
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charset = ConstructibleProjection( e )
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Display( charset )
Construct points of the moduli space (they are arrangements in all characteristics apart from $2$ and $5$):¶
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piter = PseudoIteratorOfClosedPoints( e )
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p = NextIterator( piter )
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Display( p )
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p = NextIterator( piter )
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Display( p )
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p = NextIterator( piter )
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Display( p )
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p = NextIterator( piter )
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Display( p )
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p = NextIterator( piter )
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Display( p )
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