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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 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21
      -----------------------------------------------------------------------
      -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 |

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

o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7
      -----------------------------------------------------------------------
      2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 |

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

              2              2                              2               
o16 = ideal (a  - 40b*c + 21c  + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c -
      -----------------------------------------------------------------------
         2                             2                  2                  
      50c  - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c  - 29a*d + 30b*d -
      -----------------------------------------------------------------------
                 2   2              2                              2     2  
      30c*d + 38d , b  + 21b*c + 19c  + 19a*d - 38b*d + 10c*d + 47d , b*c  +
      -----------------------------------------------------------------------
                   2         2        2       2     3   3                2   
      24b*c*d - 16c d + 39a*d  - 27b*d  + 8c*d  + 7d , c  - 22b*c*d - 24c d -
      -----------------------------------------------------------------------
           2        2        2      3
      29a*d  - 29b*d  - 10c*d  + 48d )

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  - 35b*c + 33c  + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c +
      -----------------------------------------------------------------------
         2                            2                   2                  
      33c  + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c  + 14a*d - 43b*d -
      -----------------------------------------------------------------------
                 2   2              2                     2     2            
      47c*d + 27d , b  + 22b*c - 18c  + 7b*d + 23c*d + 13d , b*c  + 34b*c*d +
      -----------------------------------------------------------------------
        2         2        2        2      3   3                2         2  
      2c d + 16a*d  + 45b*d  - 15c*d  + 10d , c  - 28b*c*d + 38c d - 39a*d  -
      -----------------------------------------------------------------------
           2        2     3
      47b*d  - 13c*d  + 4d )

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 - 16d, b + 31d, a + 12d)                                                                         |
      +----------------------------------------------------------------------------------------------------------+
      |ideal (c - 29d, b + 29d, a - 27d)                                                                         |
      +----------------------------------------------------------------------------------------------------------+
      |ideal (c + 41d, b + 35d, a - 25d)                                                                         |
      +----------------------------------------------------------------------------------------------------------+
      |                            2                     2                          2   2                      2 |
      |ideal (a - 8b - 36c + 37d, c  - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d - 9d , b  - 17b*d + 21c*d - 34d )|
      +----------------------------------------------------------------------------------------------------------+
i21 : netList decompose F2

      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                        2                              2   2              2                     2                   2                            2   2              2                              2   3                2         2        2        2     3     2               2         2        2        2      3 |
o21 = |ideal (a*c - 15b*c + 19c  + 14a*d - 43b*d - 47c*d + 27d , b  + 22b*c - 18c  + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c  + a*d + 42b*d + 37c*d + 13d , a  - 35b*c + 33c  + 46a*d - 33b*d - 48c*d + 18d , c  - 28b*c*d + 38c d - 39a*d  - 47b*d  - 13c*d  + 4d , b*c  + 34b*c*d + 2c d + 16a*d  + 45b*d  - 15c*d  + 10d )|
      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

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.