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nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

Description

The Schreyer resolution of $I$ (which is generally non-minimal) is computed. The nonminimal parts are the submatrices in this resolution which do not involve the variables in $S$. They are elements in the base ring $A$. For instance, H#(\ell, d) is the submatrix of the matrix from $C_{\ell+1} \to C_{\ell}$ sending degree $d$ to degree $d$.

The ranks of these matrices for a specific parameter value determine exactly the minimal Betti table for the ideal $I$, evaluated at that parameter point.

Now for our example.

i1 : kk = ZZ/101;
i2 : S = kk[a..d];
i3 : F = groebnerFamily ideal"a2,ab,ac,b2,bc2,c3"

             2                      2                      2               
o3 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +
                  1       3       2      4       5       6           7     
     ------------------------------------------------------------------------
                2                         2                              2  
     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  +
      9       8      10       11       12           13       15       14    
     ------------------------------------------------------------------------
                           2   2                         2                  
     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d
      16       17       18          19       21       20      22       23   
     ------------------------------------------------------------------------
           2     2                    2       2          2         2       3 
     + t  d , b*c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d ,
        24            25         27        26       28        29        30   
     ------------------------------------------------------------------------
      3                    2       2          2         2       3
     c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d )
           31         33        32       34        35        36

o3 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31
i4 : (C, H) = nonminimalMaps F;
i5 : betti(C, Weights => {1,1,1,1})

            0 1  2 3 4
o5 = total: 1 6 10 6 1
         0: 1 .  . . .
         1: . 4  4 2 .
         2: . 2  5 3 1
         3: . .  1 1 .

o5 : BettiTally

We see that there are 4 maps that are nonminimal (of sizes $2 \times 4$, $5 \times 2$, $1 \times 3$, and $1 \times 1$).

i6 : keys H

o6 = {(3, 4), (3, 5), (4, 6), (2, 3)}

o6 : List
i7 : H#(2,3)

o7 = {3} | -t_8-t_20t_13      t_7t_20-t_14t_20+t_20t_13t_19            
     {3} | -t_7+t_14-t_13t_19 -t_8-t_20t_13+t_7t_19-t_14t_19+t_13t_19^2
     ------------------------------------------------------------------------
     -t_2-t_14^2+t_20t_13^2    -t_8t_14+t_1t_20+t_7t_20t_13             |
     -t_1-2t_14t_13+t_13^2t_19 -t_2-t_7t_14-t_8t_13+t_1t_19+t_7t_13t_19 |

                                                                                                                                                                                           2                                                                                                                                                                                    4
o7 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31
i8 : H#(3,4)

o8 = {4} | -t_20                                   
     {4} | -1                                      
     {4} | t_8+t_20t_13-t_7t_19+t_14t_19-t_13t_19^2
     {4} | -t_7+t_14-t_13t_19                      
     {4} | 0                                       
     ------------------------------------------------------------------------
     -t_8                                    |
     t_13                                    |
     t_2+t_7t_14+t_8t_13-t_1t_19-t_7t_13t_19 |
     -t_1-2t_14t_13+t_13^2t_19               |
     t_7-t_14+t_13t_19                       |

                                                                                                                                                                                           5                                                                                                                                                                                    2
o8 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31
i9 : H#(3,5)

o9 = {5} | -1 t_13 -t_14 |

                                                                                                                                                                                           1                                                                                                                                                                                    3
o9 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31
i10 : H#(4,6)

o10 = {6} | -1 |

                                                                                                                                                                                            1                                                                                                                                                                                    1
o10 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

Let's impose the condition that the map H#(2,3) vanishes (so has rank 0). The Betti diagram of such ideals is not the one for a set of 6 generic points in $\PP^3$.

i11 : J = trim(minors(1, H#(2,3)) + groebnerStratum F);

o11 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ]
                   6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31
i12 : compsJ = decompose J;
i13 : #compsJ

o13 = 2
i14 : pt1 = randomPointOnRationalVariety compsJ_0

o14 = | -36 -24 -35 11 -49 -45 35 28 -36 11 -28 -30 8 -22 21 5 34 21 -10 23
      -----------------------------------------------------------------------
      27 19 -10 -30 38 -29 -8 19 39 -24 -29 -36 -16 -38 -29 24 |

               1       36
o14 : Matrix kk  <-- kk
i15 : pt2 = randomPointOnRationalVariety compsJ_1

o15 = | -20 7 11 -14 -31 20 -47 -46 -37 28 -3 -38 -23 -28 22 10 -36 16 -38
      -----------------------------------------------------------------------
      -47 -48 31 -15 -39 -2 -47 -43 -2 2 22 -18 19 0 38 -13 34 |

               1       36
o15 : Matrix kk  <-- kk
i16 : F1 = sub(F, (vars S)|pt1)

              2              2                              2               
o16 = ideal (a  + 21b*c + 11c  - 30a*d - 45b*d - 35c*d - 36d , a*b + 38b*c -
      -----------------------------------------------------------------------
         2                             2                   2                
      10c  + 27a*d + 8b*d - 36c*d - 24d , a*c - 36b*c - 29c  + 19a*d + 23b*d
      -----------------------------------------------------------------------
                   2   2              2                             2     2  
      - 22c*d - 49d , b  - 29b*c + 39c  - 16a*d + 19b*d + 5c*d + 35d , b*c  -
      -----------------------------------------------------------------------
                   2        2        2        2      3   3                2 
      24b*c*d - 10c d - 8a*d  + 34b*d  - 28c*d  + 11d , c  + 24b*c*d - 29c d
      -----------------------------------------------------------------------
             2        2        2      3
      - 38a*d  - 30b*d  + 21c*d  + 28d )

o16 : Ideal of S
i17 : betti res F1

             0 1 2 3
o17 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o17 : BettiTally
i18 : F2 = sub(F, (vars S)|pt2)

              2              2                              2              
o18 = ideal (a  + 22b*c + 28c  - 38a*d + 20b*d + 11c*d - 20d , a*b - 2b*c -
      -----------------------------------------------------------------------
         2                             2                   2                 
      38c  - 48a*d - 23b*d - 37c*d + 7d , a*c + 19b*c - 47c  - 2a*d - 47b*d -
      -----------------------------------------------------------------------
                 2   2             2                      2     2            
      28c*d - 31d , b  - 13b*c + 2c  + 31b*d + 10c*d - 47d , b*c  + 22b*c*d -
      -----------------------------------------------------------------------
         2         2        2       2      3   3                2         2  
      15c d - 43a*d  - 36b*d  - 3c*d  - 14d , c  + 34b*c*d - 18c d + 38a*d  -
      -----------------------------------------------------------------------
           2        2      3
      39b*d  + 16c*d  - 46d )

o18 : Ideal of S
i19 : betti res F2

             0 1 2 3
o19 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o19 : BettiTally

What are the ideals F1 and F2?

i20 : netList decompose F1

      +-----------------------------------------------------------------------------------------------+
o20 = |ideal (c - 49d, b - 47d, a + 42d)                                                              |
      +-----------------------------------------------------------------------------------------------+
      |ideal (c + 19d, b - 3d, a + 39d)                                                               |
      +-----------------------------------------------------------------------------------------------+
      |                                    2                      2   2      2                      2 |
      |ideal (a - 36b - 29c + 24d, b*c - 7c  - 42b*d - 49c*d - 43d , b  + 38c  + 43b*d + 39c*d - 20d )|
      +-----------------------------------------------------------------------------------------------+
i21 : netList decompose F2

      +-------------------------------------------------------+
      |                                     2              2  |
o21 = |ideal (b - 35c + 36d, a - 37c - 5d, c  + 35c*d + 44d ) |
      +-------------------------------------------------------+
      |                                      2              2 |
      |ideal (b + 46c + 46d, a + 26c + 13d, c  + 11c*d + 15d )|
      +-------------------------------------------------------+
      |                                2              2       |
      |ideal (b + 16c + 22d, a + 12d, c  - 44c*d + 27d )      |
      +-------------------------------------------------------+

We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.

See also

Ways to use nonminimalMaps:

  • nonminimalMaps(Ideal)

For the programmer

The object nonminimalMaps is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/GroebnerStrata.m2:1012:0.