Licci level Stanley Reisner ideal of CM type 2

In the paper G. Rinaldo, N. Terai, K. Yoshida, Licci level Stanley-Reisner ideals with height three and with type two, we used symbolic computation to obtain that the following three conditions are equivalent for level$^*$ squarefree monomial ideals I with codimension 3 and Cohen-Macaulay type 2:

I is licci;
The twisted conormal module of I is Cohen-Macaulay;
The second symbolic power, $I^{(2)}$, is Cohen-Macaulay.

In particular the computed-aided proofs are in Theorems 2.2, 2.3, 3.1 and 3.2 where we considered polynomial rings with 6 and 7 variables.

In short the algorithm is defined by the following 3 Steps :

Step 1: Compute all pure simplicial complexes with 6 (resp. 7) vertices whose dimension is 2 (resp. 3);
Step 2: Filter the ones produced in Step 1 having h-vector compatible to the properties in Step 3;
Step 3: Filter the ones produced in Step 2 that are level and of Cohen-Macaulay type 2 and print if the twisted conormal module and the second symbolic power are Cohen-Macaulay.

We underline that the Step 2 has been used only in the case of simplicial complexes with 7 variables to minimize the elements to check in Step 3.

In this page we describe the algorithm (Step 1, Step 2 and Step 3) and we give the results of our computation together with the source code. We refer to the article for an in-depth description of the notation and results.

Step 1. Compute all pure simplicial complexes with 6 (resp. 7) vertices whose dimension is 2 (resp. 3)

The cases with 6 and 7 vertices are similar. As an example we describe the case with 7 vertices.

A possible way to represent a pure simplicial complex on the vertices $V=\{a,b,c,d,e,f,g\}$ of dimension 3 having $f_3$ facets $F=\{E_1,\ldots, E_{f_3}\}$, is to consider a bipartite graph on the vertices $V\cup F$, that satisfies the following properties

The graph is connected;
All the vertices in $F$ have degree 4;
All the vertices in $F$ have different neighbours;
All the vertices in $V$ have at least degree 1 and at most degree $f_3-1$.

A very nice package to calculate graphs up to isomorphisms is Nauty. In this case our graph is bipartite and it needs to satisfy the 4 conditions. The command within Nauty useful for our aim is genbg with the following options

genbg -c -z $n $f  -d1:4 -D$((f-1)):4

Where

$n and $f are the cardinalities of the sets $V$ and $F$;
the option -c computes only the connected graphs (condition 1)
the option -z satisfies condition 3.
the options -d and -D satisfy conditions 2 and 4 on the degrees.

Step 2. Filter the ones in produced in Step 1 having a given h-vector

The number of pure simplicial complexes with 7 vertices produced in Step 1 is huge.

Therefore this Step is useful since we want to minimize the number of candidates that have to satisfy the two algebraic properties of Step 3, using a fast filtration.

For level ring there are nice restriction on the $h-$vector, $h=(h_0,\ldots,h_4)$. For a reference see Chapter 3, R. Stanley, Combinatorics and Commutative Algebra, 1996. In particular in this case we have that: \[ h=(1,3,h_2,h_3,2) \] with $h_3\leq h_1h_4=6$.

To compute the h-vector by a given simplicial complex it is sufficient to compute its f-vector $f_0,f_1,f_2,f_3$. That is we have to compute the number of faces of dimension 1 and 2 of the given simplicial complex. This computation is easy since we know the vertices of the facets produced in Step 1.

As an example we observe that the pure simplicial complexes with 7 vertices and 18 facets are 920443 and with the filtration of this step become 30409. For a fast implementation we used C++.

Step 3. Filter the ones produced in Step 2 that are level and of Cohen-Macaulay type 2 and print if the twisted conormal module and the second symbolic power are Cohen-Macaulay

This part of the algorithm use routine that are packages within the system for symbolic computation in Commutative Algebra Macaulay2. In particular we used the library Depth.m2 for Cohen-Macaulay recognition of a quotient ring, and the library NautyGraphs.m2 that give an interface to the files produced by Nauty.

We would like to thank Ryota Okazaki for providing the Macaulay commands to determine the Cohen-Macaulay property for the twisted conormal module of an ideal.

Implementation of the algorithm and results

In this section we give the results of our computation by the routines that we implemented in C++ (using g++ of gcc ver. 4.8) and Macaulay2 whose sources are downloadable from here.

The program in C++ has been developed under Linux on Ubuntu distribution (tested on versions 14.04 and 18.04). It has been tested on MacOs Mojave 10.14.6, too. It uses only standard library and with some effort can work on any platforms. To use it you have to download, and extract by

  tar -xzvf LicciCMtype2.tgz

move to the folder

cd LicciCMtype2

and then compile the c++ program by

make

Now we are ready to make the filtrations used in the proofs of Theorems 2.2, 2.3, 3.1 and 3.2. We show the input and the output in the console. We start with the case with 6 variables. We run the program

 
./filter6

This script implements Step1 and generate the files b9.6.nty, b10.6.nty, b11.6.nty and b12.6.nty, that contain the sets of bipartite graphs connected with the simplicial complexes with 9,10,11 and 12 facets.

Now we can filter them according to Step 3 and display the Cohen-Macaulay properties with respect to the twisted conormal module and $S/I^{(2)}$.

 
M2 CheckCM2ndSymb6.m2

The result is:

 
--------------------
f2=9
--------------------
---
(cd,abe,cf,df)
CM conormal
S/I^(2) is CM
---
(acd,be,bf,cf,adef)
CM conormal
S/I^(2) is CM
--------------------
f2=10
--------------------
---
(acd,be,cde,bf,adf,acef)
CM conormal
S/I^(2) is CM
---
(cd,abe,bf,aef)
CM conormal
S/I^(2) is CM
---
(acd,abe,bde,cf,df)
CM conormal
S/I^(2) is CM
---
(bcd,be,acde,af,cdf)
CM conormal
S/I^(2) is CM
--------------------
f2=11
--------------------
---
(cd,abe,abf,bdf,aef,cef)
CM conormal
S/I^(2) is CM
---
(acd,bcd,abe,ace,bf,def)
CM conormal
S/I^(2) is CM
--------------------
f2=12
--------------------
---
(bcd,ace,ade,bde,abf,acf,cdf,bef)
Not CM conormal

To end the computation we consider the case with 7 variables. We run the program

 
./filter7

This script implements Step1 and Step2 and generate the files b12.7.nty, b13.7.nty, b14.7.nty, b15.7.nty, b16.7.nty, b17.7.nty and b18.7.nty, that are the sets of bipartite graphs connected with the simplicial complexes with number of facets in the set $\{12,\ldots,18\}$, and that satisfy the condition on $h-$vector described in Step 2.

Now we can filter them according to Step 3 and display the Cohen-Macaulay properties with respect to the twisted conormal module and $S/I^{(2)}$.

 
M2 CheckCM2ndSymb7.m2

The result is:

 
--------------------
f3=12
--------------------
---
(abde,cf,cg,fg)
CM conormal
S/I^(2) is CM
---
(abcde,cf,adef,bg,cg)
CM conormal
S/I^(2) is CM
--------------------
f3=13
--------------------
---
(ade,bcf,cg,dg,abefg)
CM conormal
S/I^(2) is CM
--------------------
f3=14
--------------------
---
(de,cf,abdg,abeg)
CM conormal
S/I^(2) is CM
---
(abde,cdf,abef,cg,dg)
CM conormal
S/I^(2) is CM
---
(abde,abcdf,abef,cef,cg,dg)
CM conormal
S/I^(2) is CM
---
(abde,cf,cg,aeg,bdfg)
CM conormal
S/I^(2) is CM
---
(acde,cf,abdef,bg,adeg)
CM conormal
S/I^(2) is CM
--------------------
f3=15
--------------------
---
(bde,acf,cg,deg,abfg)
CM conormal
S/I^(2) is CM
---
(abcde,acf,ef,acg,bdg)
CM conormal
S/I^(2) is CM
---
(abcde,cf,bdf,abdg,aeg,ceg)
CM conormal
S/I^(2) is CM
---
(de,abcf,acef,dg,abeg,bcfg)
CM conormal
S/I^(2) is CM
---
(ade,bcf,bg,cfg)
CM conormal
S/I^(2) is CM
--------------------
f3=16
--------------------
---
(de,acf,bdg,abeg,bcfg)
CM conormal
S/I^(2) is CM
---
(abde,cf,bdg,cdg,abeg,aefg)
CM conormal
S/I^(2) is CM
---
(bde,acf,bcf,bdf,abcdg,aeg,deg)
CM conormal
S/I^(2) is CM
---
(acde,acf,bdef,bg,adeg,cfg)
CM conormal
S/I^(2) is CM
---
(bcde,abf,cf,acg,adeg,bdeg)
CM conormal
S/I^(2) is CM
---
(acde,bcf,acef,adef,bg,dfg)
CM conormal
S/I^(2) is CM
---
(ade,abdf,bef,cg)
CM conormal
S/I^(2) is CM
--------------------
f3=17
--------------------
---
(bde,acf,adef,bdg,beg,ceg)
CM conormal
S/I^(2) is CM
---
(ade,acf,def,bcg,bfg)
CM conormal
S/I^(2) is CM
---
(acde,bdf,bef,cef,abcg,adg,bdg)
CM conormal
S/I^(2) is CM
---
(abde,acf,bdf,acg,ceg,efg)
CM conormal
S/I^(2) is CM
---
(abde,acf,bcf,abef,bcg,bdg,deg)
CM conormal
S/I^(2) is CM
---
(abde,cf,bdef,abdg,bcdg,aeg)
CM conormal
S/I^(2) is CM
---
(acde,bcf,acdf,adef,bg,cdeg,aefg)
CM conormal
S/I^(2) is CM
---
(bde,acf,bdf,acg,beg,defg)
CM conormal
S/I^(2) is CM
--------------------
f3=18
--------------------
---
(ade,bcf,cdef,bcg,aceg,bfg,adfg)
CM conormal
S/I^(2) is CM
---
(ade,acf,bdef,bcg,cdeg,bfg)
CM conormal
S/I^(2) is CM
---
(bde,abcf,bef,adef,acg,cdg,adfg)
CM conormal
S/I^(2) is CM
---
(ade,cde,abcf,bcef,acg,adg,bdfg,befg)
Not CM conormal
---
(ade,abcf,abdf,bcef,cdg,beg,deg,acfg)
Not CM conormal
---
(bcde,bcf,adf,adg,aeg,bdeg,cefg)
CM conormal
S/I^(2) is CM
---
(bde,abcf,bdf,acef,acdg,beg,ceg,adfg)
Not CM conormal