# Doubling Examples for ideals of 6 points -- For an ideal $I_{\Gamma}$ of six points we compute possible doublings of $I_{\Gamma}$. See Example 2.16 in [QQ] for details

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| +-+-----------------+