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 48 -7 10 20 -23 24 16 -35 5 -36 -42 -17 21 21 29 50 -22 39 -39 12
-----------------------------------------------------------------------
19 -10 24 47 -16 -29 -29 -29 39 19 -30 -24 -38 -8 -36 |
1 36
o14 : Matrix kk <--- kk
|
i15 : pt2 = randomPointOnRationalVariety compsJ_1
o15 = | 44 39 25 16 -34 32 41 -25 9 1 -31 -2 5 38 -21 41 -47 -28 13 -39 -27
-----------------------------------------------------------------------
19 -43 34 4 2 -13 49 -47 16 -18 -47 0 22 -15 19 |
1 36
o15 : Matrix kk <--- kk
|
i16 : F1 = sub(F, (vars S)|pt1)
2 2 2
o16 = ideal (a + 21b*c + 5c - 42a*d - 23b*d - 7c*d + 43d , a*b + 47b*c +
-----------------------------------------------------------------------
2 2 2
39c + 12a*d - 17b*d - 35c*d + 48d , a*c - 30b*c - 16c - 29a*d - 39b*d
-----------------------------------------------------------------------
2 2 2 2 2
+ 21c*d + 20d , b - 8b*c - 29c - 24a*d + 19b*d + 29c*d + 24d , b*c +
-----------------------------------------------------------------------
2 2 2 2 3 3 2
39b*c*d - 10c d - 29a*d + 50b*d - 36c*d + 10d , c - 36b*c*d + 19c d
-----------------------------------------------------------------------
2 2 2 3
- 38a*d + 24b*d - 22c*d + 16d )
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 2
o18 = ideal (a - 21b*c + c - 2a*d + 32b*d + 25c*d + 44d , a*b + 4b*c + 13c
-----------------------------------------------------------------------
2 2
- 27a*d + 5b*d + 9c*d + 39d , a*c - 47b*c + 2c + 49a*d - 39b*d + 38c*d
-----------------------------------------------------------------------
2 2 2 2 2
- 34d , b - 15b*c - 47c + 19b*d + 41c*d + 41d , b*c + 16b*c*d -
-----------------------------------------------------------------------
2 2 2 2 3 3 2 2
43c d - 13a*d - 47b*d - 31c*d + 16d , c + 19b*c*d - 18c d + 22a*d
-----------------------------------------------------------------------
2 2 3
+ 34b*d - 28c*d - 25d )
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 - 29d, b + 25d, a) |
+------------------------------------------------------+
|ideal (c - 41d, b + 7d, a + 20d) |
+------------------------------------------------------+
| 2 2 |
|ideal (b + 23c + 36d, a - 33c + 31d, c - 6c*d + 41d )|
+------------------------------------------------------+
| 2 2 |
|ideal (b + 12c + 5d, a + 41c + 10d, c + 4c*d + 44d ) |
+------------------------------------------------------+
|
i21 : netList decompose F2
+---------------------------------------------------------------+
o21 = |ideal (c - 25d, b + 34d, a + 22d) |
+---------------------------------------------------------------+
| 2 2 |
|ideal (b - 6c - 18d, a + 23c - 48d, c + 20c*d + 7d ) |
+---------------------------------------------------------------+
| 3 2 2 3 |
|ideal (b - 9c + 37d, a - 17c + 37d, c - 49c d + 50c*d + 24d )|
+---------------------------------------------------------------+
|
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.
The object nonminimalMaps is a method function.