Ann Arbor, MI1992@book{alyson-thesis, author = {Reeves, A.}, title =
{Combinatorial structure on the {H}ilbert scheme}, note = {Thesis
(Ph.D.)–Cornell University}, publisher = {ProQuest LLC, Ann Arbor, MI}, year =
{1992}} ReidM.Gorenstein in codimension 4: the general structure
theoryAlgebraic geometry in east {A}sia—{T}aipei 2011Adv. Stud. Pure
Math.65201–227Math. Soc. Japan, Tokyo2015@article{ReidGeneral, author = {Reid,
M.}, title = {Gorenstein in codimension 4: the general structure theory},
booktitle = {Algebraic geometry in east {A}sia—{T}aipei 2011}, series = {Adv.
Stud. Pure Math.}, volume = {65}, pages = {201–227}, publisher = {Math. Soc.
Japan, Tokyo}, year = {2015}} RoomT.The geometry of determinantal
lociCambridge University Press1938@book{Room, author = {Room, T.}, title =
{The geometry of determinantal loci}, publisher = {Cambridge University
Press}, date = {1938}} SchenckH.StillmanM.YuanB.Calabi-yau threefolds in
${\mathbb{P}}^{n}$ and gorenstein rings,Advances in Theoretical and
Mathematical Physics, to appear@article{SSY, author = {Schenck, H.}, author =
{Stillman, M.}, author = {Yuan, B.}, title = {Calabi-Yau threefolds in $\Pn^n$
and Gorenstein rings,}, journal = {Advances in Theoretical and Mathematical
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of genus $12$Compositio Math1272001297–319@article{Schreyer, author =
{Schreyer, F.-O.}, title = {Geometry and algebra of prime Fano 3-folds of
genus $12$}, journal = {Compositio Math}, volume = {127}, year = {2001}, pages
= {297-319}} SchreyerF.-O.Die berechnung von syzygien mit dem
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title = {Die Berechnung von Syzygien mit dem verallgemeinerten
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Universit\"at Hamburg}, year = {1980}} A standard basis approach to syzygies
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@book{Voisinhodge}

  * AUTHOR = Voisin, Claire, TITLE = Hodge theory and complex algebraic geometry. I, SERIES = Cambridge Studies in Advanced Mathematics, VOLUME = 76, EDITION = English, NOTE = Translated from the French by Leila Schneps, PUBLISHER = Cambridge University Press, Cambridge, YEAR = 2007, PAGES = x+322, ISBN = 978-0-521-71801-1,

##  Appendix A Betti tables for the inverse system of a quarternary quartic

The $16$ possible Betti tables for a minimal resolution of a nondegenerate
graded Artinian Gorenstein quotient $A_{F}$ of
$S={\mathbb{C}}[x_{0},\ldots,x_{3}]$ of regularity $4$ is given in [SSY]. For
convenience we list the set of Betti tables here. They are given in macaulay2
notation, as in [GeomSyz]: if the minimal resolution is

$0\leftarrow A_{F}\leftarrow F_{0}\leftarrow F_{1}\leftarrow\ldots\leftarrow
F_{3}\leftarrow 0$

with $F_{i}=\bigoplus_{j\in{\mathbb{Z}}}b_{ij}S(-j)$ and
$b_{i}=\sum_{j}b_{ij}$, then the Betti table of an Artinian Gorenstein ring of
codimension $4$ and regularity $4$ is symmetric with the shape

$\begin{matrix}&1&b_{1}&b_{2}&b_{3}&1&\cr 0:&1&.&.&.&.&\cr
1:&.&b_{12}&b_{23}&b_{34}&.&\cr 2:&.&b_{13}&b_{24}&b_{35}&.&\cr
3:&.&b_{14}&b_{25}&b_{36}&.&\cr 4:&.&.&.&.&1&\cr\end{matrix}.$

The first eight tables are the Betti tables of arithmetic Gorenstein Calabi-
Yau threefolds in ${\mathbb{P}}^{7}$ studied by Coughlan-Golebiowski-Kapustka-
Kapustka

Table 6. Betti tables of AG Calabi-Yau threefolds in [CGKK]

Table 7. Betti tables for the remaining eight Artinian Gorenstein algebras in
[SSY]

##  Appendix B Betti table subschemes of the Hilbert scheme

In this appendix, we investigate the Betti table strata of the Hilbert scheme
$\operatorname{Hilb}=\operatorname{Hilb}^{d}({\mathbb{P}}^{3})$ whose points
parametrize subschemes ${\mathbb{V}}(I)\subset{\mathbb{P}}^{3}$ of length $d$.
We assume familiarity with term orders and Groebner bases at the level of
[CLO]. If $B$ is a Betti table the locus we wish to consider is

$\operatorname{Hilb}_{B}:=\\{[I]\in\operatorname{Hilb}\mid BT(S/I)=B\\}.$

Similarly, if $J$ is a monomial ideal, we can consider the subset

$\operatorname{Hilb}_{J}:=\\{[I]\in\operatorname{Hilb}\mid\operatorname{in}_{\text{grevlex}}(I)=J\\}.$

Both of these sets have natural structures as locally closed subschemes of
$\operatorname{Hilb}$, and for a Betti table $B$ and monomial ideal $J$, we
let

$\operatorname{Hilb}_{B,J}:=\operatorname{Hilb}_{J}\cap\operatorname{Hilb}_{B}$

We show how to use generic initial ideals, Schreyer resolutions, Groebner
strata and Macaulay2 to find the components of this subscheme of
$\operatorname{Hilb}$. In the range we are interested in ($d\leq 9$), this
locus can have more than one component, but each component turns out to be
rational for these degrees. The computations involve ideals in lots of
variables (basically, affine coordinates of open subsets of
$\operatorname{Hilb}$), so we sometimes need to use special features to get
computations to finish.

In this paper, we are interested in the $\operatorname{Hilb}_{B}$, where $B$
is the Betti table of an ideal with $4\leq d\leq 9$ points, and every ideal
with the given Betti table has $\operatorname{reg}(S/I)\leq 2$. In this
appendix, we will explain how we investigate the components of these
$\operatorname{Hilb}_{B}$, focusing on the case $d=6$.

We are interested in the case when the characteristic of the base field
${\mathbb{K}}$ is zero. Most of our computations will be done over a finite
field, though, but with large enough characteristic so that the results will
hold in characteristic zero.

###  B.1. Strongly stable ideals with a given Hilbert polynomial

A strongly stable monomial ideal is a monomial ideal $J\subset
S={\mathbb{K}}[x_{0},\ldots,x_{n}]$ such that if $x_{i}m$ is in $J$, then
$x_{j}m$ is in $J$, for all $j<i$.

In characteristic zero (or more generally, if the characteristic is greater
than the regularity of $J$), this coincides with the notion of an ideal being
fixed under the Borel action of upper triangular matrices.

There are only finitely many saturated strongly stable monomial ideals of a
given Hilbert polynomial (this is folklore, but D. Maclagan has a nice general
paper [maclagan] which proves a sweeping generalization of this). Alyson
Reeves [alyson-thesis] gave an algorithm in her Cornell thesis to compute all
of the strongly stable ideals, with a given Hilbert polynomial. Moore and
Nagel [moore-nagel], and separately Albarelli-Lella [alb-lella] have papers
(and the latter has a Macaulay2 implementation improving this construction).

We compute using Macaulay2, the case when $d=6$. We use the
StronglyStableIdeals Macaulay2 package written by Albarelli and Lella.

    i2 : needsPackage "QuartenaryQuartics";
    i3 : kk = ZZ/32003;
    i4 : S = kk[a..d];
    i5 : B6 = stronglyStableIdeals(6, S);

    i6 : netList B6

         +----------------------------------+
         |              6                   |
    o6 = |ideal (b, a, c )                  |
         +----------------------------------+
         |                2   5             |
         |ideal (a, b*c, b , c )            |
         +----------------------------------+
         |           2     2   4            |
         |ideal (a, b , b*c , c )           |
         +----------------------------------+
         |           3     2   2    3       |
         |ideal (a, c , b*c , b c, b )      |
         +----------------------------------+
         |                  2        2   4  |
         |ideal (b*c, a*c, b , a*b, a , c ) |
         +----------------------------------+
         |             2        2   3     2 |
         |ideal (a*c, b , a*b, a , c , b*c )|
         +----------------------------------+

    i7 : netList pack(3, B6/minimalBetti)

         +--------------+--------------+--------------+
         |       0 1 2 3|       0 1 2 3|       0 1 2 3|
    o7 = |total: 1 3 3 1|total: 1 4 5 2|total: 1 4 5 2|
         |    0: 1 2 1 .|    0: 1 1 . .|    0: 1 1 . .|
         |    1: . . . .|    1: . 2 3 1|    1: . 1 1 .|
         |    2: . . . .|    2: . . . .|    2: . 1 2 1|
         |    3: . . . .|    3: . . . .|    3: . 1 2 1|
         |    4: . . . .|    4: . 1 2 1|              |
         |    5: . 1 2 1|              |              |
         +--------------+--------------+--------------+
         |       0 1 2 3|       0 1 2 3|       0 1 2 3|
         |total: 1 5 7 3|total: 1 6 8 3|total: 1 6 8 3|
         |    0: 1 1 . .|    0: 1 . . .|    0: 1 . . .|
         |    1: . . . .|    1: . 5 6 2|    1: . 4 4 1|
         |    2: . 4 7 3|    2: . . . .|    2: . 2 4 2|
         |              |    3: . 1 2 1|              |
         +--------------+--------------+--------------+

Thus, there are six saturated strongly stable monomial ideals, although four
of these are degenerate: they lie on a hyperplane. Of the remaining two, only
one has regularity $\operatorname{reg}(S/J)=2$.

###  B.2. Generic initial ideals

There are two basic theorems (see Eisenbud’s commutative algebra book
[CAwvtAG], for an exposition) that we will use:

######  Theorem B.1 (Galligo [Ga74], Bayer-Stillman [BSDuke]).

Fix a term order $>$, and let ${\mathbb{K}}$ be an infinite field. Let
$I\subset T={\mathbb{K}}[x_{0},\ldots,x_{n}]$ be a homogeneous ideal. After a
general change of coordinates $g\in GL(n+1,{\mathbb{K}})$, the initial ideal
$J=in_{>}(g\cdot I)$ in this term order is Borel fixed.

If the term order is the graded reverse lexicographic order, we call this
monomial ideal the generic initial ideal of $I$ (or $\operatorname{gin}(I)$).

######  Theorem B.2 (Bayer-Stillman [BS87]).

The generic initial ideal $J=\operatorname{gin}(I)$ has the same depth,
projective dimension, and regularity as $I$.

In particular, after a general, or sufficiently random, change of coordinates,
the initial ideal $J$ is saturated if $I$ is, it is Borel fixed, and has the
same regularity and projective dimension as $I$.

In characteristic zero (or if the characteristic is greater than the largest
degree generator of $J$), $J$ is strongly stable, and the regularity of $I$ is
precisely the maximal degree of a generator of $J$.

In the range of degrees we are discussing, for characteristic larger than,
say, 100, strongly stable ideals and Borel fixed ideals coincide, so we do not
encounter the subtleties that occur in finite characteristic.

Thus, we can partition the set of ideals $I$ with $[I]\in H$ by their generic
initial ideal $J=\operatorname{gin}(I)$.

In our running example, there is exactly one strongly stable saturated ideal
$J$ with no linear forms in the ideal, and $\operatorname{reg}(S/J)\leq 2$.

    i8 : select(B6, J -> first degree J_0 != 1 and regularity (S^1/J) <= 2)

                       2        2   3     2
    o8 = {ideal (a*c, b , a*b, a , c , b*c )}

    i9 : J = ideal(a^2, a*b, b^2, a*c, b*c^2, c^3)

                 2        2          2   3
    o9 = ideal (a , a*b, b , a*c, b*c , c )

    i10 : betti res J

                 0 1 2 3
    o10 = total: 1 6 8 3
              0: 1 . . .
              1: . 4 4 1
              2: . 2 4 2

We compare this to the resolution of the ideal $I$ of 6 general (random)
points in ${\mathbb{P}}^{3}$. A general fact (which follows from upper-
semicontinuity) is that the Betti table of $I$ is entry by entry at most the
Betti table of its initial ideal $J$.

    i11 : I = pointsIdeal randomPoints(S, 6);

    o11 : Ideal of S

    i12 : J == ideal leadTerm I

    o12 = true

    i13 : (betti res I, betti res J)

                  0 1 2 3         0 1 2 3
    o13 = (total: 1 4 5 2, total: 1 6 8 3)
               0: 1 . . .      0: 1 . . .
               1: . 4 2 .      1: . 4 4 1
               2: . . 3 2      2: . 2 4 2

###  B.3. The Groebner family and the Groebner stratum of an initial ideal

Given a monomial ideal $J$, and a term order (in this paper, we always take
the graded reverse lex order), we can form a parameter space of all ideals
having initial ideal $J$. This gives rise to a locally closed subscheme
$\operatorname{Hilb}_{J}\subset\operatorname{Hilb}$ of the Hilbert scheme,
whose points correspond to ideals having initial ideal $J$. See Lella-Roggero
[lella-roggero] for additional details.

We now sketch the construction. For each minimal generator $x^{\alpha}$ of
$J$, consider the set $x^{\gamma}$ of monomials of the same degree, not in
$J$. For each such $\alpha$, form a polynomial

$F_{\alpha}=x^{\alpha}+\sum\limits_{\gamma<\alpha}t_{\alpha,\gamma}x^{\gamma}$

Let $U=T[x_{0},\ldots,x_{n}]$, where $T={\mathbb{K}}[t_{\alpha,\gamma}]$ is
the polynomial ring generated by all variables of the form $t_{\alpha,\gamma}$
as above. Let $F$ be the ideal in $U$ generated by the $F_{\alpha}$. The ideal
$F=(F_{\alpha})$ is called the Groebner family of $J$.

Any term order $>$ on the monomials of $S$, when restricted to a finite set
$M$ of monomials, is given by a weight vector $w\in\mathbb{Z}^{n+1}$: for
$x^{\alpha},x^{\beta}\in M$, $x^{\alpha}>x^{\beta}$ if and only if
$w\cdot\alpha>w\cdot\beta$. If we let the weight of $t_{\alpha,\gamma}$ be
$w\cdot(\alpha-\gamma)$, then with this grading, $F_{\alpha}$ is homogeneous
of weight $w\cdot\alpha$. This grading is useful for performance reasons:
computing with homogeneous ideals is much more efficient than with arbitrary
ideals. Additionally, we can order the variables $t_{\alpha,\gamma}$ by
refining the weight degree by any term order on $T$: if the weight of
$t_{\alpha,\gamma}$ is greater than $t_{\alpha^{\prime},\gamma^{\prime}}$ then
$t_{\alpha,\gamma}>t_{\alpha^{\prime},\gamma^{\prime}}$ in this term order.
This insures that if a homogeneous polynomial in $T$ has a term which is a
variable, then the lead term will be a variable.

We want the ideal $L\subset T$ in the parameter variables that encodes the
constraints that insure $F$ is a Groebner basis. Let $L$ be the ideal
generated by all of the coefficients in $T$ of the reductions of the
S-polynomials on the $F_{\alpha}$. The ideal $L$ parametrizes all ideals which
have S-pairs reducing to zero. The scheme ${\mathbb{V}}(L)$ with ideal $L$ is
called the Groebner stratum of $J$.

Over ${\mathbb{V}}(L)$, $F$ defines a flat family all of whose fibers have
initial ideal $J$, and consequently defines via representability of the
Hilbert scheme, a locally closed subscheme $Hilb_{J}:={\mathbb{V}}(L)$ of
$\operatorname{Hilb}$. These families, over all saturated monomial ideals,
cover the Hilbert scheme. For $p\in{\mathbb{V}}(L)$, let $F_{p}\subset S$ be
the ideal corresponding to that point.

If we restrict to generic initial ideals, then every ideal is equivalent,
under a change of coordinates, to an ideal corresponding to a point in one of
these subsets of $\operatorname{Hilb}$.

In our running example above, this Groebner stratum is an open affine subset
of the Hilbert scheme. As we saw earlier (and also follows directly),
$\dim\operatorname{Hilb}=18$, and is known to be irreducible and rational. We
check the dimension and irreducibility of $\operatorname{Hilb}_{J}$, which is
an open subset of $\operatorname{Hilb}$.

    i14 : F = groebnerFamily J;
    i15 : U = ring F;
    i16 : T = coefficientRing U;

    i17 : netList F_*

          +--------------------------------------------------------------+
          | 2                      2                      2              |
    o17 = |a  + t b*c + t a*d + t c  + t b*d + t c*d + t d               |
          |      1       3       2      4       5       6                |
          +--------------------------------------------------------------+
          |                         2                         2          |
          |a*b + t b*c + t a*d + t c  + t  b*d + t  c*d + t  d           |
          |       7       9       8      10       11       12            |
          +--------------------------------------------------------------+
          | 2                         2                         2        |
          |b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d         |
          |      13       15       14      16       17       18          |
          +--------------------------------------------------------------+
          |                            2                         2       |
          |a*c + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d        |
          |       19       21       20      22       23       24         |
          +--------------------------------------------------------------+
          |   2                    2       2          2         2       3|
          |b*c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d |
          |        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    |
          +--------------------------------------------------------------+

    i18 : L = trim groebnerStratum F;

    o18 : Ideal of T

    i19 : dim L == 18

    o19 = true

    i20 : isPrime L

    o20 = true

###  B.4. The Schreyer resolution and minimal Betti numbers

Schreyer’s algorithm for computing free resolutions [Schreyer80] can be used
to construct equations for $\operatorname{Hilb}_{B,J}$ from the Groebner
family and stratum.

For any Betti table $B$, and saturated monomial ideal $J$, we obtain the
locally closed subscheme $\operatorname{Hilb}_{B,J}$, consisting of all
$p\in{\mathbb{V}}(L)$ such that $F_{p}$ has initial ideal $J$, and $S/F_{p}$
has Betti table $B$. The steps for this construction are as follows.

  1. (1)

A free resolution $C$ of the ideal $F$ over $U/L$ can be computed via
Schreyer’s algorithm [Schreyer80, lascala].

  2. (2)

This resolution restricts to a minimal free resolution of the initial ideal
$J$ if $J$ is strongly stable.

  3. (3)

We get a number of degree 0 maps in this resolution, say $\phi_{i,d}$, with
entries in the ring $T/L$. We consider the loci of these maps with prescribed
ranks. The ranks of these maps describe where cancellation in the resolution
can occur, and consequently describe the Betti table. For fixed size of minors
of each of these matrices, we get a subscheme of ${\mathbb{V}}(L)$ which is
the union of all of the $\operatorname{Hilb}_{B,J}$, over all Betti tables
entrywise greater than the one corresponding to the rank conditions.

  4. (4)

To be a bit more precise, for any point $p\in{\mathbb{V}}(L)\subset Hilb$, we
get a nonminimal resolution $C$ for $S/F_{p}$. The minimal Betti table of this
ideal can be computed from $C$ by

$b_{id}(S/F_{p})=\operatorname{rk}(C_{i})-\operatorname{rk}(\phi_{i,d})(p)-\operatorname{rk}(\phi_{i-1,d})(p).$

Similarly, given a Betti table $B$ which is entrywise $\leq B_{J}$, there
exists a unique assignment of ranks which correspond to this Betti table.

  5. (5)

Thus, given a strongly stable ideal $J$ and a Betti table $B$, we can consider
the locus $\overline{\operatorname{Hilb}}_{B,J}\subset\operatorname{Hilb}$ in
${\mathbb{V}}(L)$, defined by the minors of these $\phi_{i,d}$ of the
corresponding size. This is a closed subscheme of ${\mathbb{V}}(L)$ containing
the union of all $\operatorname{Hilb}_{B^{\prime},J}$, with $B^{\prime}\geq
B$.

It turns out for small numbers of points, we can compute all of these loci!
Rather, we can compute enough about them to identify the irreducible
components and to show that they are rational.

A complication with doing this in Macaulay2 arises, but it is a small issue:
we can lift a free resolution of $J$ to one of $F$, but it is not always the
minimal resolution of $J$. Still, we can use this to determine the equations
of the $\overline{\operatorname{Hilb}}_{B,J}$ locally on ${\mathbb{V}}(L)$,
and consequently we can determine the component structure of the
$\operatorname{Hilb}_{B,J}$

We now illustrate the above technique in our running example. We have written
a function in Macaulay2, in the package ParameterSchemes, which computes this
lifted resolution, and the various degree 0 maps whose ranks determine the
Betti table of the minimal free resolution (at any point of
${\mathbb{V}}(L)$).

    i21 : (CF, H) = nonminimalMaps F;
    i22 : U = ring CF;

    i23 : CF

           1      6      10      6      1
    o23 = U  <-- U  <-- U   <-- U  <-- U

          0      1      2       3      4

    o23 : ChainComplex

    i24 : betti(CF, Weights=>{1}) -- in this case it is indeed a non-minimal resolution

                 0 1  2 3 4
    o24 = total: 1 6 10 6 1
              0: 1 .  . . .
              1: . 4  4 2 .
              2: . 2  5 3 1
              3: . .  1 1 .

    i25 : isHomogeneous CF -- but it is homogeneous, as it needs to be.
    o25 = true

Notice that the resulting complex is not minimal when restricted to the origin
(i.e. the point $J$). The following 4 matrices determine the minimal Betti
numbers of points on $\operatorname{Hilb}$. In fact, the last 2 matrices
always have full rank (as they must), and the $5\times 2$ matrix $M_{2}$
always has rank at least 1.

    i26 : keys H -- these are the maps of scalars in the resolution.
                 -- Key is (homological degree, internal degree).
    o26 = {(3, 4), (3, 5), (4, 6), (2, 3)}

    o26 : List

    i27 :  M1 = H#(2,3) -- rank is 0, 1, or 2.

    o27 = {3} | -t_8-t_14t_19      t_7t_14-t_20t_14+t_14t_19t_13
          {3} | -t_7+t_20-t_19t_13 -t_8-t_14t_19+t_7t_13-t_20t_13+t_19t_13^2
          ---------------------------------------------------------------------
          -t_2-t_20^2+t_14t_19^2    -t_8t_20+t_1t_14+t_7t_14t_19             |
          -t_1-2t_20t_19+t_19^2t_13 -t_2-t_7t_20-t_8t_19+t_1t_13+t_7t_19t_13 |

                  2       4
    o27 : Matrix T  <--- T

    i28 : M2 = H#(3,4) -- rank is 1 or 2

    o28 = {4} | -t_14
          {4} | -1
          {4} | t_8+t_14t_19-t_7t_13+t_20t_13-t_19t_13^2
          {4} | -t_7+t_20-t_19t_13
          {4} | 0
          ---------------------------------------------------------------------
          -t_8                                    |
          t_19                                    |
          t_2+t_7t_20+t_8t_19-t_1t_13-t_7t_19t_13 |
          -t_1-2t_20t_19+t_19^2t_13               |
          t_7-t_20+t_19t_13                       |

                  5       2
    o28 : Matrix T  <--- T

    i29 : M3 = H#(3,5) -- maximal rank, can ignore

    o29 = {5} | -1 t_19 -t_20 |

                  1       3
    o29 : Matrix T  <--- T

    i30 : M4 = H#(4,6) -- maximal rank, can ignore

    o30 = {6} | -1 |

                  1       1
    o30 : Matrix T  <--- T

    i31 : minors(2, M2) == minors(1, M1)

    o31 = true

Since the ideal of maximal minors of $M_{2}$ is equal to the ideal of the
maximal minors of $M_{1}$ (which are the entries), we obtain 3 possible rank
conditions

$(\operatorname{rk}(M_{1}),\operatorname{rk}(M_{2}))\in\\{(0,1),(1,2),(2,2)\\}.$

These correspond to three Betti table loci $\operatorname{Hilb}_{B,J}$. The
generic case (ranks are maximal) corresponds to the Betti table of 6 general
points in ${\mathbb{P}}^{3}$. The minimal rank case corresponds to the Betti
table of $J$. All three Betti tables occur as cases (5, 6, 7) in Table 1.

###  B.5. The components of the Betti table loci of degree $6$

We will next compute the components of the
$\overline{\operatorname{Hilb}}_{B,J}$. We can compute the loci in
${\mathbb{V}}(L)$ that correspond to these three rank conditions. This will
lead us to the following loci:

  1. (1)

6 general points. This locus is dense in $\operatorname{Hilb}$, which is
rational of dimension 18, and has Betti table type (5) in Table 1.

  2. (2)

3 points on a line, and 3 general points. The closure of the locus of such
points is rational and irreducible, and has dimension 16, and has Betti table
type (6) in Table 1.

  3. (3)

5 points in a plane and one point off the plane. The closure of the locus of
such points is rational and irreducible, and has dimension 16, and has Betti
table type (7) in Table 1.

  4. (4)

3 points on each of 2 skew lines. The closure of the locus of such points is
rational and irreducible, and has dimension 14, and has Betti table type (7)
in Table 1.

We now explain how to determine this information. Above, we have already
handled case (1): the ideal $L$ itself is prime of dimension 18, and is dense
in the Hilbert scheme $\operatorname{Hilb}$ parametrizing length 6 subschemes
of ${\mathbb{P}}^{3}$.

We compute the ideals $L_{441}$ and $L_{430}$ defining the rank loci
$\overline{\operatorname{Hilb}}_{B,J}$. Recall that the corresponding Betti
tables are:

$\small\begin{matrix}&0&1&2&3\\\ \text{total:}&1&6&8&3\\\
\text{0:}&1&\text{.}&\text{.}&\text{.}\\\ \text{1:}&\text{.}&4&4&1\\\
\text{2:}&\text{.}&2&4&2\\\ \end{matrix}\qquad\begin{matrix}&0&1&2&3\\\
\text{total:}&1&5&6&2\\\ \text{0:}&1&\text{.}&\text{.}&\text{.}\\\
\text{1:}&\text{.}&4&3&\text{.}\\\ \text{2:}&\text{.}&1&3&2\\\ \end{matrix}$

The first step is to compute the ideals $L_{441}$ and $L_{430}$ which define
the rank loci, and decompose these loci into irreducible components.

    i32 : L441 = trim(minors(1, M1) + L);
    o32 : Ideal of T
    i33 : L430 = trim(minors(2, M1) + L);
    o33 : Ideal of T
    i34 : compsL441 = decompose L441;
    i35 : compsL430 = decompose L430;
    i36 : (#compsL441, #compsL430)

    o36 = (2, 2)

    i37 : compsL441/dim

    o37 = {16, 14}

    i38 : compsL430/dim

    o38 = {16, 16}

    i39 : compsL441_0 == compsL430_0
    o39 = true

    i40 : isSubset(compsL430_1, compsL441_1)
    o40 = true

The last 2 lines confirm that the rank locus ${\mathbb{V}}(L_{430})$ contains
the rank locus ${\mathbb{V}}(L_{441})$. We see that there are 2 irreducible
components of $\operatorname{Hilb}_{B_{441},J}$ of dimensions 14 and 16, and
that there is one irreducible component of $\operatorname{Hilb}_{B_{430},J}$,
it has dimension 16.

Next, we check that the 2 components for the $441$ locus are both rational.
The function randomPointOnRationalVariety returns a ${\mathbb{K}}$-valued
point, if certain sufficient conditions for the variety to be rational are
satisfied. This holds for both components of $L_{441}$, so they are rational,
of dimensions 14 and 16.

We now take the random point on each (rational) component that this function
returns, and compute the corresponding ideal in $S$. By considering its
irreducible decomposition, we obtain a candidate for the general element on
each component. In these cases, a dimension count matches the dimension we
computed for the component.

We implement this with the code below. The dimension 14 component has general
element 3 points on one line, and 3 points on another skew line. The dimension
16 component has general element 5 points in a plane, and one point off the
plane.

    i42 : pta = randomPointOnRationalVariety compsL441_0

    o42 = | 4818 14868 14185 11709 -12680 -3804 11646 9165 1329 -11430 -14770
          ---------------------------------------------------------------------
          -11245 -7240 13564 8800 -14581 12746 -792 15900 4173 15035 1560 -5149
          ---------------------------------------------------------------------
          12775 4571 7802 1805 13264 -10075 7936 -805 1321 -2281 9132 10444
          ---------------------------------------------------------------------
          3905 |

                   1        36
    o42 : Matrix kk  <--- kk

    i43 : ptb = randomPointOnRationalVariety compsL441_1

    o43 = | -10329 -120 -8822 7364 -103 11106 -435 -15249 -10586 -3570 -12139
          ---------------------------------------------------------------------
          6555 -10648 -8190 5354 1013 -5960 -67 13303 11931 -5834 13202 10214
          ---------------------------------------------------------------------
          10956 -2813 -12260 14265 8927 12815 -7346 11947 -23 0 11453 -5155
          ---------------------------------------------------------------------
          -3006 |

                   1        36
    o43 : Matrix kk  <--- kk

    i44 : I441a = sub(F, (vars S) | pta)

                  2                   2
    o44 = ideal (a  + 8800b*c - 11430c  - 11245a*d - 3804b*d + ..., ...)

    o44 : Ideal of S

    i45 : I441b = sub(F, (vars S) | ptb)

                  2                  2
    o45 = ideal (a  + 5354b*c - 3570c  + 6555a*d + ..., ...)

    o45 : Ideal of S

    i46 : betti res I441a

                 0 1 2 3
    o46 = total: 1 6 8 3
              0: 1 . . .
              1: . 4 4 1
              2: . 2 4 2

    o46 : BettiTally

    i47 : betti res I441b

                 0 1 2 3
    o47 = total: 1 6 8 3
              0: 1 . . .
              1: . 4 4 1
              2: . 2 4 2

    o47 : BettiTally

    i48 : decompose I441a

    o48 = {ideal (c - 10075d, b + 10116d, a + 15005d), ideal (a + 1321b + 7802c
          ---------------------------------------------------------------------
                     2                    2                             2   3
          - 12657d, b  + 10444b*c + 13264c  + 6479b*d - 11887c*d + 7735d , c  +
          ---------------------------------------------------------------------
                          2            2          2        3     2
          3905b*c*d - 805c d + 14534b*d  - 9978c*d  - 1947d , b*c  + 7936b*c*d
          ---------------------------------------------------------------------
                 2           2           2        3
          - 5149c d - 3437b*d  + 15943c*d  + 7452d )}

    o48 : List

    i49 : (decompose I441a)/degree

    o49 = {1, 5}

Finally, the ideal $L_{430}$ does not meet the sufficient conditions to
determine that it is rational. The workaround is to find a rational family of
ideals which each have Betti table $B_{430}$, and form a family of dimension
16. Since ${\mathbb{V}}(L_{430})$ is irreducible of dimension 16 and must
contain this family, they must coincide, and therefore that component is
rational of dimension 16. The general element is a set of 3 points on a line,
together with 3 general points in space.

The computation below yields the Betti table $B_{430}$ for a random such
ideal.

    i50 : I430 = pointsIdeal randomBlockMatrix({S^2, S^2}, {S^3, S^3},
              {{random, random},
               {0, random}})
                                                2   2
    o50 = ideal (a*d - 9045b*d - 9460c*d + 2639d , c  + ..., ...)

    o50 : Ideal of S

    i51 : betti res I430

                 0 1 2 3
    o51 = total: 1 5 6 2
              0: 1 . . .
              1: . 4 3 .
              2: . 1 3 2

This vignette illustrates the techniques to analyze the case of 6 points. For
degrees $4\leq d\leq 9$, a similar analysis allows us to produce all of the
components and show that they are rational.