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 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
-----------------------------------------------------------------------
-36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
1 36
o14 : Matrix kk <--- kk
|
i15 : pt2 = randomPointOnRationalVariety compsJ_1
o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
-----------------------------------------------------------------------
-35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
1 36
o15 : Matrix kk <--- kk
|
i16 : F1 = sub(F, (vars S)|pt1)
2 2 2
o16 = ideal (a - 44b*c - 35c + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
-----------------------------------------------------------------------
2 2 2
15c - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c - 16a*d + 2b*d +
-----------------------------------------------------------------------
2 2 2 2 2
19c*d + 34d , b - 30b*c + 19c - 22a*d + 21b*d + 50c*d - 12d , b*c -
-----------------------------------------------------------------------
2 2 2 2 3 3 2
29b*c*d - 36c d + 24a*d + 2b*d + 50c*d + 9d , c - 24b*c*d + 39c d -
-----------------------------------------------------------------------
2 2 2 3
8a*d - 29b*d - 29c*d - 17d )
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 + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
-----------------------------------------------------------------------
2 2 2
45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d
-----------------------------------------------------------------------
2 2 2 2 2
- 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , b*c - 43b*c*d +
-----------------------------------------------------------------------
2 2 2 2 3 3 2 2
19c d + 34a*d + 21b*d + 46c*d + 20d , c + 38b*c*d + 22c d - 15a*d
-----------------------------------------------------------------------
2 2 3
- 47b*d - 39c*d + 40d )
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 - 13d, b + 32d, a + 36d) |
+------------------------------------------------------+
|ideal (c - 16d, b + d, a + 16d) |
+------------------------------------------------------+
| 2 2 |
|ideal (b - 6c + 33d, a - 36c + 2d, c + 43c*d - d ) |
+------------------------------------------------------+
| 2 2 |
|ideal (b + 29c + 7d, a - 19c + 24d, c - 20c*d - 30d )|
+------------------------------------------------------+
|
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 + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - 47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d )|
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|
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.