If $\Gamma$ does not contain four collinear points, then $S/I_{\Gamma}$ has regularity $2$. The following computation shows that a random element of $Hom(\omega(-4),R/I_{\Gamma})$ is injective. Therefore Corollary 2.15 gives the corresponding doublings.
i1 : kk = ZZ/101; |
i2 : R = kk[x_0..x_3]; |
i3 : HT = bettiStrataExamples R; |
i4 : netList for k in {"[420]","[430]","[441a]","[441b]"} list (
if doubling(8,pointsIdeal((HT#k)_0))===null then
{k, betti res pointsIdeal((HT#k)_0), "No injective map"}
else
{k, betti res pointsIdeal((HT#k)_0),
betti res doubling(8,pointsIdeal((HT#k)_0))}
)
+------+--------------+-----------------+
| | 0 1 2 3| 0 1 2 3 4|
o4 = |[420] |total: 1 4 5 2|total: 1 6 10 6 1|
| | 0: 1 . . .| 0: 1 . . . .|
| | 1: . 4 2 .| 1: . 4 2 . .|
| | 2: . . 3 2| 2: . 2 6 2 .|
| | | 3: . . 2 4 .|
| | | 4: . . . . 1|
+------+--------------+-----------------+
| | 0 1 2 3| 0 1 2 3 4|
|[430] |total: 1 5 6 2|total: 1 7 12 7 1|
| | 0: 1 . . .| 0: 1 . . . .|
| | 1: . 4 3 .| 1: . 4 3 . .|
| | 2: . 1 3 2| 2: . 3 6 3 .|
| | | 3: . . 3 4 .|
| | | 4: . . . . 1|
+------+--------------+-----------------+
| | 0 1 2 3| 0 1 2 3 4|
|[441a]|total: 1 6 8 3|total: 1 9 16 9 1|
| | 0: 1 . . .| 0: 1 . . . .|
| | 1: . 4 4 1| 1: . 4 4 1 .|
| | 2: . 2 4 2| 2: . 4 8 4 .|
| | | 3: . 1 4 4 .|
| | | 4: . . . . 1|
+------+--------------+-----------------+
| | 0 1 2 3| 0 1 2 3 4|
|[441b]|total: 1 6 8 3|total: 1 9 16 9 1|
| | 0: 1 . . .| 0: 1 . . . .|
| | 1: . 4 4 1| 1: . 4 4 1 .|
| | 2: . 2 4 2| 2: . 4 8 4 .|
| | | 3: . 1 4 4 .|
| | | 4: . . . . 1|
+------+--------------+-----------------+
|
Next, suppose $\Gamma$ is the set of six points span $\mathbb{P}^{3}$ and four are collinear. We first check that a random element of $Hom(\omega(-\gamma), R/I_{\Gamma})$ is not injective for $\gamma = 2$. When $\gamma\geq 3$, a general element is injective and we compute the Betti table of doubling of $I_{\Gamma}$ with a general element for $\gamma=3,4,5,6$.
i5 : Mpts = randomPoints(R,4,2)|(randomPoints(R,2,4)||(randomPoints(R,2,4)*0));
4 6
o5 : Matrix R <--- R
|
i6 : IGamma = pointsIdeal(Mpts); o6 : Ideal of R |
i7 : betti res IGamma
0 1 2 3
o7 = total: 1 6 8 3
0: 1 . . .
1: . 5 6 2
2: . . . .
3: . 1 2 1
o7 : BettiTally
|
i8 : netList for k in {2,3,4,5,6} list (
if doubling(k+4,IGamma)===null then {k, "No injective map"}
else {k, betti res doubling(k+4,IGamma)})
+-+-----------------+
o8 = |2|No injective map |
+-+-----------------+
| | 0 1 2 3 4|
|3|total: 1 6 10 6 1|
| | 0: 1 1 . . .|
| | 1: . 3 5 2 .|
| | 2: . 2 5 3 .|
| | 3: . . . 1 1|
+-+-----------------+
| | 0 1 2 3 4|
|4|total: 1 9 16 9 1|
| | 0: 1 . . . .|
| | 1: . 6 8 3 .|
| | 2: . . . . .|
| | 3: . 3 8 6 .|
| | 4: . . . . 1|
+-+-----------------+
| | 0 1 2 3 4|
|5|total: 1 9 16 9 1|
| | 0: 1 . . . .|
| | 1: . 5 6 2 .|
| | 2: . 1 2 1 .|
| | 3: . 1 2 1 .|
| | 4: . 2 6 5 .|
| | 5: . . . . 1|
+-+-----------------+
| | 0 1 2 3 4|
|6|total: 1 9 16 9 1|
| | 0: 1 . . . .|
| | 1: . 5 6 2 .|
| | 2: . . . . .|
| | 3: . 2 4 2 .|
| | 4: . . . . .|
| | 5: . 2 6 5 .|
| | 6: . . . . 1|
+-+-----------------+
|