Licci level Stanley Reisner ideal of height 3 and dimension at most 4

In the paper G. Rinaldo, N. Terai, Licci level Stanley-Reisner ideals with height three, we used symbolic computation to obtain that the following two conditions are equivalent for level$^*$ squarefree monomial ideals I with codimension 3 and dimension 4:

I is licci;
The twisted conormal module of I is Cohen-Macaulay;

Moreover we observed that when the 2 conditions hold the second symbolic power, $I^{(2)}$, is Cohen-Macaulay.

In particular the computed-aided proofs are in Proposition 2.1, 2.2, Theorems 3.1, and Theorem 4.1 where we considered polynomial rings with 5, 6 and 7 variables.

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

Step 1: Compute all pure simplicial complexes with 5, (resp. 6, resp. 7) vertices whose dimension is 1 (resp. 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 print if the twisted conormal module are Cohen-Macaulay. When this is the case check that the second symbolic power is 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 $n$ vertices whose dimension is given

The cases with 5, 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: \[ h=(1,3,h_2,h_3,h_4) \] with $h_3\leq h_1h_4$, and $h_4$ is the Cohen-Macaulay type, that in this case is induced by the Euler characteristic.

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.

Step 3. Filter the ones produced in Step 2 that are level and of a given Cohen-Macaulay type 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 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 LicciCMtype345.tgz

move to the folder

cd LicciCMtype345

and then compile the c++ program by

make

Now we are ready to make the filtrations used in the proofs. We show the input and the output in the console. We start with the case with 5 variables. The case with 6 variables is similar. We run the program

 
./filter5

This script implements Step1 and generate the files b5.5.nty, b6.5.nty, b7.5.nty, b8.5.nty, b9.5.nty and b10.5.nty, that contain the sets of bipartite graphs connected with the simplicial complexes with $f$ facets and $f\in \{5,\ldots, 10\}$.

Now we can filter them according to Step 3 and display the Cohen-Macaulay properties with respect to the twisted conormal module and $S/I^{(2)}$. For example we show the ones having 5 vertices and 7 facets by

 
M2 --script CheckCM2ndSymb.m2 5 7

The result is:

 
CM type=3
(abc,ad,bcd,be,ce)
CM conormal!
S/I^(2) is CM
----------
CM type=3
(bc,abd,acd,be,ce,ade)
Not CM conormal
S/I^(2) is CM
----------
CM type=3
(bc,abd,acd,ae,ce,bde)
Not CM conormal
S/I^(2) is CM

We observe that $(abc,ad,bcd,be,ce)$ is the only monomial ideal of Cohen-Macaulay type 3 that is licci.

Now, as ane example we compute the case with 7 variables of Cohen-Macaulay type 3 with number of facets within the interval $[15,20]$. We run the program (the computation needs more or less 15 minutes on a laptop with an Intel core i5)

 
./filter7 3 15 20

This script implements Step1 and Step2 and generate the files b15.7.3.nty, b16.7.3.nty, b17.7.3.nty, b18.7.3.nty, b19.7.3.nty, and b20.7.3.nty, that are the sets of bipartite graphs related to the simplicial complexes with number of facets in the set $\{15,\ldots,20\}$, and that satisfy the condition on $h-$vector described in Step 2.

Now we can filter, as an example, the ones with 15 facets using the algorithm defined in Step 3 and display the Cohen-Macaulay properties with respect to the twisted conormal module and $S/I^{(2)}$.

 
M2 --script CheckCM2ndSymb.m2 7 15 3

The result is:

 
----------
CM type=3
(acdg,abceg,bde,bf,ef)
CM conormal!
S/I^(2) is CM
----------
CM type=3
(acdg,bce,abdeg,bf,cf,adefg)
Not CM conormal
S/I^(2) is CM
----------
CM type=3
(abcdg,bce,adeg,cf,df,abefg)
Not CM conormal
S/I^(2) is CM
----------
CM type=3
(abcdg,be,acdeg,bf,dfg,acef)
Not CM conormal
S/I^(2) is CM
----------
CM type=3
(bcdg,ae,af,bfg,cdefg,bcdef)
CM conormal!
S/I^(2) is CM
----------
CM type=3
(abcdg,be,acdeg,af,cdfg)
CM conormal!
S/I^(2) is CM

There are in fact 3 that are licci within this set as proved in the paper.

Step 1. Compute all pure simplicial complexes with \(n\) vertices whose dimension is given

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

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

Implementation of the algorithm and results