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Realizations of self-projecting matroids

This collection contains the (self-projecting) realization spaces of self-projecting matroids of rank k on n elements over characteristic zero for (k,n) in {(2,4),...,(2,12),(3,6),(3,7),(3,8),(4,8),(4,9),(5,10)}. It accompanies the article "The self-projecting Grassmannian" by Alheydis Geiger and Francesca Zaffalon [GZ25].

Warning: The database is still under construction. The collections for selfprojecting matroids of rank 4 on 9 elements and for selfprojecting matroids of rank 5 on 10 elements are not complete yet, but underway.

For the cases {(2,4),(3,6),(4,8),(5,10)} the database stores material from the article [GHSV24], for which the accompanying code (in Macaulay2, Magma, Matlab, OSCAR and SageMath) can be found on github.

How to access the database

After installing OSCAR, you can access the database as described here https://docs.oscar-system.org/dev/Experimental/OscarDB/introduction/#get_db The realization spaces of the matroids are contained in the collection $Combinatorics.SelfProjectingMatroids$

You can query the database using the following parameters

  • identifier of the database entry $data.name$
  • rank of the matroid $data.rank$
  • size of the groundset of the matroid $data.length_groundset$
  • dimension of its realization space $data.dim_r$
  • dimension of its self-projecting realization space $data.dim_s$
  • whether the realization space and the self-projecting realization space are equal $data.equality_of_realizationspaces$

Note that all query entries in the according dictionaries are strings, except if the value asked for is $nothing$.

Warning: for rank 3 on 8 elements, for rank 4 on 9 elements and for rank 5 on 10 elements the computation of the selfprojecting realization space did not always terminate. In these cases (as in the example above) the proeprties that could not be computed, like$dim_s$, $equality_of_realizationspaces$ and $selfprojecting_realization_space$, are set to $nothing$.

julia> r3n8 = find_one(db["Combinatorics.SelfProjectingMatroids"], Dict(["data.rank"=>"3", "data.length_groundset"=>"8", "data.dim_s"=>nothing]))
The matroid is of rank 3 on 8 elements.
The realization space is
  [1   0   0   1       1      1    x[4]    x[5]]
  [0   1   0   1   x[12]   x[8]       1       1]
  [0   0   1   1   x[12]      1   x[15]   x[15]]
in the multivariate polynomial ring in 15 variables over QQ
within the vanishing set of the ideal
Ideal with 10 generators
avoiding the zero loci of the polynomials
RingElem[x[12], x[15], -x[8], -x[8] + 1, x[15] - 1, x[4], x[5], -x[12] + 1, x[4] - x[15], x[5] - x[15], -x[4] + 1, -x[5] + 1, x[8] - x[12], x[4] - x[5], x[8]*x[15] - 1, x[4]*x[12] - x[15], x[5]*x[12] - x[15], -x[4]*x[12] + 1, -x[5]*x[12] + 1, -x[4]*x[8] + 1, -x[5]*x[8] + 1, -x[4]*x[8]*x[12] + x[4]*x[12] + x[8]*x[15] - x[12]*x[15] + x[12] - 1, -x[5]*x[8]*x[12] + x[5]*x[12] + x[8]*x[15] - x[12]*x[15] + x[12] - 1]
The computation of the self-projecting realization space did not terminate.

Once you have decided on the database entry you want to investigate more closely you have the following options.

julia> MR = find_one(db["Combinatorics.SelfProjectingMatroids"], Dict(["data.rank"=>"4","data.length_groundset"=>"9","data.equality_of_realizationspaces"=>"false"]))
The matroid is of rank 4 on 9 elements.
The realization space is
  [1   0   0   0   1   x[3]    x[3]       1       1]
  [0   1   0   0   1   x[9]       1    x[9]       1]
  [0   0   1   0   1      1   x[13]   x[14]   x[15]]
  [0   0   0   1   1      1       1       1       0]
in the multivariate polynomial ring in 20 variables over QQ
within the vanishing set of the ideal
Ideal with 15 generators
avoiding the zero loci of the polynomials
RingElem[-x[13], -x[14], -x[15], -x[13] + 1, -x[14] + 1, x[13] - x[14], x[9], x[9] - 1, -x[9] + x[14], x[15] - 1, -x[13] + x[15], -x[3], -x[3] + 1, x[3] - x[13], x[14] - x[15], -x[3] + x[9], -x[3] - x[9] + 2, x[9]*x[13] - 1, x[9]*x[15] - 1, -x[9]*x[13] + x[14], x[9]*x[15] - x[14], x[9]*x[15] - x[14] - x[15] + 1, -x[9]*x[15] - x[13] + x[15] + 1, x[9]*x[15] + x[13] - x[14] - x[15], -x[3]*x[14] + 1, -x[3]*x[15] + 1, -x[3]*x[14] + x[13], -x[3]*x[15] + x[13], -x[3]*x[15] + x[13] + x[15] - 1, x[3]*x[15] + x[14] - x[15] - 1, x[3]*x[15] - x[13] + x[14] - x[15], x[3]*x[9] - 1, x[3]*x[9] + x[3]*x[13] - 2*x[3] - x[9]*x[13] + 1, -x[3]*x[9] + x[3]*x[14] - x[9]*x[14] + 2*x[9] - 1, -x[3]*x[9] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + 1, x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - x[3]*x[9] - x[3]*x[14] - x[9]*x[13] + 1, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] + 1, -x[3]*x[9]*x[15] + x[3]*x[14] - x[9]*x[14] + x[9]*x[15] + x[9] - 1, -x[3]*x[9]*x[15] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + x[15], x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - 2*x[3]*x[9] - x[3]*x[13] - x[3]*x[14] + 2*x[3] - x[9]*x[13] - x[9]*x[14] + 2*x[9] + x[13] + x[14] - 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] - x[9]*x[15] + x[9] + x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] - x[9]*x[14] + x[9]*x[15] + x[9] - x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] + x[9]*x[13] + x[9]*x[15] - x[9] - x[13] - x[14] - x[15] + 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] + x[9]*x[14] - x[9]*x[15] - x[9] - x[13] - x[14] + x[15] + 2]
The selfprojecting realization space is
  [1   0   0   0   1   x[3]    x[3]       1       1]
  [0   1   0   0   1   x[9]       1    x[9]       1]
  [0   0   1   0   1      1   x[13]   x[14]   x[15]]
  [0   0   0   1   1      1       1       1       0]
in the multivariate polynomial ring in 20 variables over QQ
within the vanishing set of the ideal
Ideal with 24 generators
avoiding the zero loci of the polynomials
RingElem[-x[13], -x[14], -x[15], -x[13] + 1, -x[14] + 1, x[13] - x[14], x[9], x[9] - 1, -x[9] + x[14], x[15] - 1, -x[13] + x[15], -x[3], -x[3] + 1, x[3] - x[13], x[14] - x[15], -x[3] + x[9], -x[3] - x[9] + 2, x[9]*x[13] - 1, x[9]*x[15] - 1, -x[9]*x[13] + x[14], x[9]*x[15] - x[14], x[9]*x[15] - x[14] - x[15] + 1, -x[9]*x[15] - x[13] + x[15] + 1, x[9]*x[15] + x[13] - x[14] - x[15], -x[3]*x[14] + 1, -x[3]*x[15] + 1, -x[3]*x[14] + x[13], -x[3]*x[15] + x[13], -x[3]*x[15] + x[13] + x[15] - 1, x[3]*x[15] + x[14] - x[15] - 1, x[3]*x[15] - x[13] + x[14] - x[15], x[3]*x[9] - 1, x[3]*x[9] + x[3]*x[13] - 2*x[3] - x[9]*x[13] + 1, -x[3]*x[9] + x[3]*x[14] - x[9]*x[14] + 2*x[9] - 1, -x[3]*x[9] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + 1, x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - x[3]*x[9] - x[3]*x[14] - x[9]*x[13] + 1, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] + 1, -x[3]*x[9]*x[15] + x[3]*x[14] - x[9]*x[14] + x[9]*x[15] + x[9] - 1, -x[3]*x[9]*x[15] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + x[15], x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - 2*x[3]*x[9] - x[3]*x[13] - x[3]*x[14] + 2*x[3] - x[9]*x[13] - x[9]*x[14] + 2*x[9] + x[13] + x[14] - 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] - x[9]*x[15] + x[9] + x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] - x[9]*x[14] + x[9]*x[15] + x[9] - x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] + x[9]*x[13] + x[9]*x[15] - x[9] - x[13] - x[14] - x[15] + 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] + x[9]*x[14] - x[9]*x[15] - x[9] - x[13] - x[14] + x[15] + 2]
The closures of the realization space and the self-projecting realization space are not equal.

julia> name(MR)
"r_4_n_9_0002"

julia> Oscar.matroid(MR)
Matroid of rank 4 on 9 elements

julia> rank(MR)
4

julia> length_groundset(MR)
9

julia> realization_space(MR)
The realization space is
  [1   0   0   0   1   x[3]    x[3]       1       1]
  [0   1   0   0   1   x[9]       1    x[9]       1]
  [0   0   1   0   1      1   x[13]   x[14]   x[15]]
  [0   0   0   1   1      1       1       1       0]
in the multivariate polynomial ring in 20 variables over QQ
within the vanishing set of the ideal
Ideal with 15 generators
avoiding the zero loci of the polynomials
RingElem[-x[13], -x[14], -x[15], -x[13] + 1, -x[14] + 1, x[13] - x[14], x[9], x[9] - 1, -x[9] + x[14], x[15] - 1, -x[13] + x[15], -x[3], -x[3] + 1, x[3] - x[13], x[14] - x[15], -x[3] + x[9], -x[3] - x[9] + 2, x[9]*x[13] - 1, x[9]*x[15] - 1, -x[9]*x[13] + x[14], x[9]*x[15] - x[14], x[9]*x[15] - x[14] - x[15] + 1, -x[9]*x[15] - x[13] + x[15] + 1, x[9]*x[15] + x[13] - x[14] - x[15], -x[3]*x[14] + 1, -x[3]*x[15] + 1, -x[3]*x[14] + x[13], -x[3]*x[15] + x[13], -x[3]*x[15] + x[13] + x[15] - 1, x[3]*x[15] + x[14] - x[15] - 1, x[3]*x[15] - x[13] + x[14] - x[15], x[3]*x[9] - 1, x[3]*x[9] + x[3]*x[13] - 2*x[3] - x[9]*x[13] + 1, -x[3]*x[9] + x[3]*x[14] - x[9]*x[14] + 2*x[9] - 1, -x[3]*x[9] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + 1, x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - x[3]*x[9] - x[3]*x[14] - x[9]*x[13] + 1, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] + 1, -x[3]*x[9]*x[15] + x[3]*x[14] - x[9]*x[14] + x[9]*x[15] + x[9] - 1, -x[3]*x[9]*x[15] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + x[15], x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - 2*x[3]*x[9] - x[3]*x[13] - x[3]*x[14] + 2*x[3] - x[9]*x[13] - x[9]*x[14] + 2*x[9] + x[13] + x[14] - 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] - x[9]*x[15] + x[9] + x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] - x[9]*x[14] + x[9]*x[15] + x[9] - x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] + x[9]*x[13] + x[9]*x[15] - x[9] - x[13] - x[14] - x[15] + 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] + x[9]*x[14] - x[9]*x[15] - x[9] - x[13] - x[14] + x[15] + 2]

julia> dim_r(MR)
5

julia> selfprojecting_realization_space(MR)
The selfprojecting realization space is
  [1   0   0   0   1   x[3]    x[3]       1       1]
  [0   1   0   0   1   x[9]       1    x[9]       1]
  [0   0   1   0   1      1   x[13]   x[14]   x[15]]
  [0   0   0   1   1      1       1       1       0]
in the multivariate polynomial ring in 20 variables over QQ
within the vanishing set of the ideal
Ideal with 24 generators
avoiding the zero loci of the polynomials
RingElem[-x[13], -x[14], -x[15], -x[13] + 1, -x[14] + 1, x[13] - x[14], x[9], x[9] - 1, -x[9] + x[14], x[15] - 1, -x[13] + x[15], -x[3], -x[3] + 1, x[3] - x[13], x[14] - x[15], -x[3] + x[9], -x[3] - x[9] + 2, x[9]*x[13] - 1, x[9]*x[15] - 1, -x[9]*x[13] + x[14], x[9]*x[15] - x[14], x[9]*x[15] - x[14] - x[15] + 1, -x[9]*x[15] - x[13] + x[15] + 1, x[9]*x[15] + x[13] - x[14] - x[15], -x[3]*x[14] + 1, -x[3]*x[15] + 1, -x[3]*x[14] + x[13], -x[3]*x[15] + x[13], -x[3]*x[15] + x[13] + x[15] - 1, x[3]*x[15] + x[14] - x[15] - 1, x[3]*x[15] - x[13] + x[14] - x[15], x[3]*x[9] - 1, x[3]*x[9] + x[3]*x[13] - 2*x[3] - x[9]*x[13] + 1, -x[3]*x[9] + x[3]*x[14] - x[9]*x[14] + 2*x[9] - 1, -x[3]*x[9] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + 1, x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - x[3]*x[9] - x[3]*x[14] - x[9]*x[13] + 1, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] + 1, -x[3]*x[9]*x[15] + x[3]*x[14] - x[9]*x[14] + x[9]*x[15] + x[9] - 1, -x[3]*x[9]*x[15] + x[3]*x[14] + x[9]*x[13] - x[13] - x[14] + x[15], x[3]*x[9]*x[13] + x[3]*x[9]*x[14] - 2*x[3]*x[9] - x[3]*x[13] - x[3]*x[14] + 2*x[3] - x[9]*x[13] - x[9]*x[14] + 2*x[9] + x[13] + x[14] - 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] - x[9]*x[13] - x[9]*x[15] + x[9] + x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] - x[9]*x[14] + x[9]*x[15] + x[9] - x[15], -x[3]*x[9]*x[15] + x[3]*x[14] + x[3]*x[15] - x[3] + x[9]*x[13] + x[9]*x[15] - x[9] - x[13] - x[14] - x[15] + 2, x[3]*x[9]*x[15] + x[3]*x[13] - x[3]*x[15] - x[3] + x[9]*x[14] - x[9]*x[15] - x[9] - x[13] - x[14] + x[15] + 2]

julia> dim_s(MR)
3

julia> equality_of_realizationspaces(MR)
false

The realization space $\mathcal{R}$ obtained by $realization_space(MR)$ and the self-projecting realization space $\mathcal{S}$ obtained by $selfprojecting_realization_space(MR)$ can be investigated using the code in the experimental section of OSCAR on MatroidRealizationSpaces.

julia> R = realization_space(MR);

julia> defining_ideal(R)
Ideal generated by
  x[1] - 1
  x[2] - x[3]
  x[4] - 1
  x[5] - 1
  x[6] - 1
  x[7] - x[9]
  x[8] - 1
  x[10] - 1
  x[11] - 1
  x[12] - 1
  x[16] - 1
  x[17] - 1
  x[18] - 1
  x[19] - 1
  x[20]

julia> ambient_ring(R)
Multivariate polynomial ring in 20 variables x[1], x[2], x[3], x[4], ..., x[20]
  over rational field
  
julia> S = selfprojecting_realization_space(MR);

julia> defining_ideal(S)
Ideal generated by
  x[1] - 1
  x[2] - x[3]
  x[3]^2*x[9]*x[13] - x[3]^2*x[9] - x[3]^2*x[13]*x[14] + x[3]^2*x[14] - x[3]*x[9
  ]^2*x[14] + x[3]*x[9]^2 - 2*x[3]*x[9]*x[13] + 2*x[3]*x[9]*x[14] + 2*x[3]*x[13]
  *x[14] - 3*x[3]*x[14] + x[3] + x[9]^2*x[13]*x[14] - x[9]^2*x[13] - 2*x[9]*x[13
  ]*x[14] + 3*x[9]*x[13] - x[9] - x[13] + x[14]
  x[3]^2*x[9]*x[15] - x[3]^2*x[14]*x[15] + x[3]*x[9]*x[14] - x[3]*x[9]*x[15]^2 -
   x[3]*x[9]*x[15] - x[3]*x[9] - x[3]*x[13]*x[14]*x[15] + x[3]*x[13]*x[14] + x[3
  ]*x[14]*x[15]^2 + x[3]*x[14]*x[15] - x[3]*x[14] + x[3]*x[15] - x[9]*x[13]*x[14
  ]*x[15] + x[9]*x[13]*x[15]^2 + x[9]*x[15] + x[13]*x[14] - x[13] - x[14] - x[15
  ]^2 + 1
  x[3]^2*x[13]*x[14]*x[15] - x[3]^2*x[13]*x[14] - x[3]^2*x[14]^2*x[15] + x[3]^2*
  x[14]*x[15] + x[3]^2*x[14] - x[3]^2*x[15] + x[3]*x[9]*x[14]^2 - x[3]*x[9]*x[14
  ] - x[3]*x[9]*x[15]^2 + x[3]*x[9]*x[15] + x[3]*x[13]*x[14]^2 - 3*x[3]*x[13]*x[
  14]*x[15] + x[3]*x[13]*x[14] + x[3]*x[13] + x[3]*x[14]^2*x[15] - x[3]*x[14]^2
  + x[3]*x[14]*x[15]^2 - 2*x[3]*x[14] + x[3]*x[15] - x[9]*x[13]*x[14]^2 + x[9]*x
  [13]*x[14] + x[9]*x[13]*x[15]^2 - x[9]*x[13]*x[15] + x[13]*x[14] + x[13]*x[15]
   - 2*x[13] - x[14]*x[15] + x[14] - x[15]^2 + x[15]
  x[3]*x[9]^2*x[15] + x[3]*x[9]*x[13] - x[3]*x[9]*x[15]^2 - x[3]*x[9]*x[15] - x[
  3]*x[9] - x[3]*x[13]*x[14]*x[15] + x[3]*x[14]*x[15]^2 + x[3]*x[15] - x[9]^2*x[
  13]*x[15] - x[9]*x[13]*x[14]*x[15] + x[9]*x[13]*x[14] + x[9]*x[13]*x[15]^2 + x
  [9]*x[13]*x[15] - x[9]*x[13] + x[9]*x[15] + x[13]*x[14] - x[13] - x[14] - x[15
  ]^2 + 1
  x[3]*x[9]*x[13]^2 - x[3]*x[9]*x[13] - x[3]*x[9]*x[15]^2 + x[3]*x[9]*x[15] - x[
  3]*x[13]^2*x[14] + x[3]*x[13]*x[14] + x[3]*x[14]*x[15]^2 - x[3]*x[14]*x[15] -
  x[9]^2*x[13]^2*x[15] + x[9]^2*x[13]*x[14]*x[15] - x[9]^2*x[13]*x[14] + x[9]^2*
  x[13]*x[15] + x[9]^2*x[13] - x[9]^2*x[15] + x[9]*x[13]^2*x[14] + x[9]*x[13]^2*
  x[15] - x[9]*x[13]^2 - 3*x[9]*x[13]*x[14]*x[15] + x[9]*x[13]*x[14] + x[9]*x[13
  ]*x[15]^2 - 2*x[9]*x[13] + x[9]*x[14] + x[9]*x[15] + x[13]*x[14] - x[13]*x[15]
   + x[13] + x[14]*x[15] - 2*x[14] - x[15]^2 + x[15]
  x[3]*x[9]*x[13]*x[15] - x[3]*x[9]*x[15]^2 - x[3]*x[13]*x[14]*x[15] + x[3]*x[14
  ]*x[15]^2 - x[9]*x[13]*x[14]*x[15] + x[9]*x[13]*x[14] + x[9]*x[13]*x[15]^2 - x
  [9]*x[13]*x[15] - x[9]*x[13] + x[9]*x[15] + x[13]*x[14] - x[14] - x[15]^2 + x[
  15]
  x[3]*x[9]*x[14]*x[15] - x[3]*x[9]*x[15]^2 - x[3]*x[13]*x[14]*x[15] + x[3]*x[13
  ]*x[14] + x[3]*x[14]*x[15]^2 - x[3]*x[14]*x[15] - x[3]*x[14] + x[3]*x[15] - x[
  9]*x[13]*x[14]*x[15] + x[9]*x[13]*x[15]^2 + x[13]*x[14] - x[13] - x[15]^2 + x[
  15]
  x[3]*x[13]^2*x[14]*x[15] - x[3]*x[13]^2*x[14] - x[3]*x[13]*x[14]^2*x[15] - x[3
  ]*x[13]*x[14]*x[15]^2 + 2*x[3]*x[13]*x[14]*x[15] + x[3]*x[13]*x[14] - x[3]*x[1
  3]*x[15] + x[3]*x[14]^2*x[15]^2 - x[3]*x[14]*x[15]^2 - x[3]*x[14]*x[15] + x[3]
  *x[15]^2 + x[9]*x[13]^2*x[14]*x[15] - x[9]*x[13]^2*x[15]^2 - x[9]*x[13]*x[14]^
  2*x[15] + x[9]*x[13]*x[14]^2 + x[9]*x[13]*x[14]*x[15]^2 - 2*x[9]*x[13]*x[14]*x
  [15] - x[9]*x[13]*x[14] + x[9]*x[13]*x[15]^2 + x[9]*x[13]*x[15] + x[9]*x[14]*x
  [15] - x[9]*x[15]^2 - x[13]^2*x[14] + x[13]^2 + x[13]*x[14]^2 + x[13]*x[15]^2
  - 2*x[13]*x[15] - x[14]^2 - x[14]*x[15]^2 + 2*x[14]*x[15]
  x[4] - 1
  x[5] - 1
  x[6] - 1
  x[7] - x[9]
  x[8] - 1
  x[9]^2*x[13]^2*x[15]^2 - x[9]^2*x[13]*x[14]*x[15]^2 + x[9]^2*x[13]*x[14]*x[15]
   - x[9]^2*x[13]*x[15]^2 - x[9]^2*x[13]*x[15] + x[9]^2*x[15]^2 - 2*x[9]*x[13]^2
  *x[14]*x[15] + x[9]*x[13]^2*x[14] - x[9]*x[13]^2 + 2*x[9]*x[13]*x[14]*x[15]^2
  + x[9]*x[13]*x[14]*x[15] - x[9]*x[13]*x[14] - 2*x[9]*x[13]*x[15]^2 + 3*x[9]*x[
  13]*x[15] + x[9]*x[13] - x[9]*x[14]*x[15] - x[9]*x[15] + x[13]^2*x[14] - 2*x[1
  3]*x[14] - x[14]*x[15]^2 + x[14]*x[15] + x[14] + x[15]^2 - x[15]
  x[10] - 1
  x[11] - 1
  x[12] - 1
  x[16] - 1
  x[17] - 1
  x[18] - 1
  x[19] - 1
  x[20]

The example above showcased $\mathcal{R}\supsetneq\mathcal{S}$. Below you see an example with $\mathcal{R}=\mathcal{S}$.

julia> MR = find_one(db["Combinatorics.SelfProjectingMatroids"], Dict("data.name"=>"r_3_n_8_10"))
The matroid is of rank 3 on 8 elements.
The realization space is
  [1   0   0   1       1   x[14]    x[4]       1]
  [0   1   0   1   x[12]       1       1       1]
  [0   0   1   1   x[12]   x[14]   x[14]   x[15]]
in the multivariate polynomial ring in 15 variables over QQ
within the vanishing set of the ideal
Ideal with 11 generators
avoiding the zero loci of the polynomials
RingElem[x[12], x[14], x[15], x[14] - 1, x[15] - 1, -x[14] + x[15], x[4], -x[12] + 1, x[4] - x[14], x[12] - x[15], -x[4] + 1, x[4]*x[12] - x[14], -x[4]*x[15] + x[14], -x[12]*x[14] + 1, -x[4]*x[12] + 1, -x[12]*x[14]*x[15] + 2*x[12]*x[14] - x[12] - x[14] + x[15], -x[4]*x[12]*x[15] + x[4]*x[12] + x[12]*x[14] - x[12] - x[14] + x[15]]
The selfprojecting realization space is
  [1   0   0   1       1   x[14]    x[4]       1]
  [0   1   0   1   x[12]       1       1       1]
  [0   0   1   1   x[12]   x[14]   x[14]   x[15]]
in the multivariate polynomial ring in 15 variables over QQ
within the vanishing set of the ideal
Ideal with 11 generators
avoiding the zero loci of the polynomials
RingElem[x[12], x[14], x[15], x[14] - 1, x[15] - 1, -x[14] + x[15], x[4], -x[12] + 1, x[4] - x[14], x[12] - x[15], -x[4] + 1, x[4]*x[12] - x[14], -x[4]*x[15] + x[14], -x[12]*x[14] + 1, -x[4]*x[12] + 1, -x[12]*x[14]*x[15] + 2*x[12]*x[14] - x[12] - x[14] + x[15], -x[4]*x[12]*x[15] + x[4]*x[12] + x[12]*x[14] - x[12] - x[14] + x[15]]
The closures of the realization space and the self-projecting realization space are equal.

julia> name(MR)
"r_3_n_8_10"

julia> Oscar.matroid(MR)
Matroid of rank 3 on 8 elements

julia> rank(MR)
3

julia> length_groundset(MR)
8

julia> realization_space(MR)
The realization space is
  [1   0   0   1       1   x[14]    x[4]       1]
  [0   1   0   1   x[12]       1       1       1]
  [0   0   1   1   x[12]   x[14]   x[14]   x[15]]
in the multivariate polynomial ring in 15 variables over QQ
within the vanishing set of the ideal
Ideal with 11 generators
avoiding the zero loci of the polynomials
RingElem[x[12], x[14], x[15], x[14] - 1, x[15] - 1, -x[14] + x[15], x[4], -x[12] + 1, x[4] - x[14], x[12] - x[15], -x[4] + 1, x[4]*x[12] - x[14], -x[4]*x[15] + x[14], -x[12]*x[14] + 1, -x[4]*x[12] + 1, -x[12]*x[14]*x[15] + 2*x[12]*x[14] - x[12] - x[14] + x[15], -x[4]*x[12]*x[15] + x[4]*x[12] + x[12]*x[14] - x[12] - x[14] + x[15]]

julia> dim_r(MR)
4

julia> selfprojecting_realization_space(MR)
The selfprojecting realization space is
  [1   0   0   1       1   x[14]    x[4]       1]
  [0   1   0   1   x[12]       1       1       1]
  [0   0   1   1   x[12]   x[14]   x[14]   x[15]]
in the multivariate polynomial ring in 15 variables over QQ
within the vanishing set of the ideal
Ideal with 11 generators
avoiding the zero loci of the polynomials
RingElem[x[12], x[14], x[15], x[14] - 1, x[15] - 1, -x[14] + x[15], x[4], -x[12] + 1, x[4] - x[14], x[12] - x[15], -x[4] + 1, x[4]*x[12] - x[14], -x[4]*x[15] + x[14], -x[12]*x[14] + 1, -x[4]*x[12] + 1, -x[12]*x[14]*x[15] + 2*x[12]*x[14] - x[12] - x[14] + x[15], -x[4]*x[12]*x[15] + x[4]*x[12] + x[12]*x[14] - x[12] - x[14] + x[15]]

julia> dim_s(MR)
4

julia> equality_of_realizationspaces(MR)
true

How to verify claims from the article

To verify Tables 2, 3, and 4 from the article, one can use queries to the database. The example below shows how to generate the line of Table 2 with respect to the matroids of rank 3 on 7 elements. Recall that the uniform matroids are not stored in the database. The other rows as well as Table 3 can be verified similarly. Note that the database collection for (4,9) is not filled completely yet.

julia> t2 = [length(db["Combinatorics.SelfProjectingMatroids"], Dict("data.rank"=>"3", "data.length_groundset"=>"7","data.dim_s"=>"$i")) for i in -1:5]
7-element Vector{Int64}:
 1
 1
 1
 3
 3
 1
 1

To obtain numbers from Table 4, i.e. the distribution of realizable matroids without selfprojecting realization you can use the following code:

julia> t4 = [length(db["Combinatorics.SelfProjectingMatroids"], Dict("data.rank"=>"4", "data.length_groundset"=>"9","data.dim_r"=>"$i","data.dim_s"=>"-1")) for i in 0:6]
7-element Vector{Int64}:
    4
  103
  494
 1089
  738
  124
    4

One can count the matroids for which $\mathcal{R}$ and $\mathcal{S}$ are known and do not coincide as follows:

julia> length(db["Combinatorics.SelfProjectingMatroids"], Dict("data.rank"=>"4", "data.length_groundset"=>"9","data.equality_of_realizationspaces"=>"false"))
5399

Theorem 4.11 claims that there are at least 5400 matroids of rank 4 on 9 elements with $\mathcal{R}\supsetneq\mathcal{S}$. Since the database does not count the uniform matroid $U_{4,9}$, the claim is verified.

The user can find the selfprojecting matroids for which the computation of the selfprojecting realization space was too costly and did not terminate as follows:

julia> notterminated = length([r for r in find(db["Combinatorics.SelfProjectingMatroids"], Dict(["data.rank"=>"3", "data.length_groundset"=>"8","data.dim_s"=>nothing]))])
4

Example 4.12

In order to work with the database and/or compute self-projecting realization spaces of matroids in OSCAR, you need to use version 1.6.0 or later. To reproduce example 4.12 you can access the relevant file from the database.

julia> using Oscar

julia> db = Oscar.OscarDB.get_db();

julia> find_one(db["Combinatorics.SelfProjectingMatroids"], Dict(["name"=>"r_4_n_9_index_5985"]))
The matroid is of rank 4 on 9 elements.
The realization space is
  [1   0   0   0   2//3   0      1   1   1//2]
  [0   1   0   0      0   2   1//2   1   1//2]
  [0   0   1   0      1   1      1   1      1]
  [0   0   0   1      2   2      2   1      1]
in the multivariate polynomial ring in 20 variables over QQ
within the vanishing set of the ideal
Ideal with 20 generators
avoiding the zero loci of the polynomials
RingElem[2]
The matroid does not have a self-projecting realization over characteristic zero.
The closures of the realization space and the self-projecting realization space are not equal.

Additional Code

Additional code as well as the original code and output from the computations in magma can be found at https://github.com/AlheydisGeiger/selfprojectingGrassmannian

Contact

Please direct questions about this part of OSCAR to the following people:

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.

Software used to create the database collection $Combinatorics.SelfProjectingMatroids$: Magma (V2.27), Julia (Version 1.12.1), OSCAR (version 1.6.0-DEV), GNU parallel 20221122

Last updated 11/12/2025.