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GroebnerStrata :: nonminimalMaps

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

Synopsis

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 = | 47 23 27 -45 2 -2 35 -24 -5 -5 35 26 -48 -10 -32 9 8 24 -33 -11 25
      -----------------------------------------------------------------------
      -38 21 -8 -41 19 39 -24 -29 19 -29 -29 -30 -36 -16 -22 |

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

o15 = | 2 15 28 -14 38 31 26 -39 24 28 -12 48 30 -13 -46 -7 -36 -47 27 -47 34
      -----------------------------------------------------------------------
      -21 16 -43 38 -18 22 6 19 -39 38 -28 0 34 2 -15 |

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

              2             2                             2               
o16 = ideal (a  - 32b*c - 5c  + 26a*d - 2b*d + 27c*d + 47d , a*b - 41b*c -
      -----------------------------------------------------------------------
         2                             2                   2                
      33c  + 25a*d - 48b*d - 5c*d + 23d , a*c - 29b*c + 19c  - 24a*d - 11b*d
      -----------------------------------------------------------------------
                  2   2              2                             2     2  
      - 10c*d + 2d , b  - 16b*c - 29c  - 30a*d - 38b*d + 9c*d + 35d , b*c  +
      -----------------------------------------------------------------------
                   2         2       2        2      3   3                2 
      19b*c*d + 21c d + 39a*d  + 8b*d  + 35c*d  - 45d , c  - 22b*c*d - 29c d
      -----------------------------------------------------------------------
             2       2        2      3
      - 36a*d  - 8b*d  + 24c*d  - 24d )

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  - 46b*c + 28c  + 48a*d + 31b*d + 28c*d + 2d , a*b + 38b*c +
      -----------------------------------------------------------------------
         2                              2                   2               
      27c  + 34a*d + 30b*d + 24c*d + 15d , a*c - 28b*c - 18c  + 6a*d - 47b*d
      -----------------------------------------------------------------------
                   2   2             2                     2     2          
      - 13c*d + 38d , b  + 2b*c + 19c  - 21b*d - 7c*d + 26d , b*c  - 39b*c*d
      -----------------------------------------------------------------------
           2         2        2        2      3   3                2   
      + 16c d + 22a*d  - 36b*d  - 12c*d  - 14d , c  - 15b*c*d + 38c d +
      -----------------------------------------------------------------------
           2        2        2      3
      34a*d  - 43b*d  - 47c*d  - 39d )

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 - 2d, b + 45d, a - 29d)                       |
      +-------------------------------------------------------+
      |ideal (c - 24d, b - 14d, a + 15d)                      |
      +-------------------------------------------------------+
      |                                      2              2 |
      |ideal (b - 31c - 16d, a + 29c - 18d, c  - 40c*d - 11d )|
      +-------------------------------------------------------+
      |                                    2             2    |
      |ideal (b + c - 31d, a + 48c - 49d, c  + 32c*d + 3d )   |
      +-------------------------------------------------------+
i21 : netList decompose F2

      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                      2              2                                                                                                                                                                |
o21 = |ideal (b - 47c - 10d, a - 44c + 10d, c  + 34c*d - 17d )                                                                                                                                                               |
      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |        2                              2                                 2                                   2   2                              2                                2   2                              2 |
      |ideal (c  + 46a*d - 26b*d - 35c*d + 23d , b*c - 3a*d - 37b*d + 47c*d + 8d , a*c + 43a*d - 36b*d - 34c*d - 31d , b  + 41a*d + 42b*d - 42c*d - 23d , a*b + 17a*d + 17b*d - 9c*d - d , a  + 36a*d - 34b*d + 39c*d + 29d )|
      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

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 :

For the programmer

The object nonminimalMaps is a method function.