Let $\mathcal{L}$ be the cycle space of the complete graph on five nodes, $K_5$, viewed as a linear subspace in $\mathbb{P}^9$. Its reciprocal $\mathcal{L}^{-1}$ is an irreducible variety in $\mathbb{P}^9$. Find a Gröbner basis for its homogenous prime ideal.

One has that

$$\mathcal{L}=\ker \begin{pmatrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\ -1 & 0 & 0 & 0 1 & 1 &1 & 0 &0 & 0\\ 0 & -1 & 0 & 0 &-1 & 0 & 0 &1 &1 & 0 \\ 0 & 0 &-1 & 0 & 0 &-1 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & -1 & 0 & 0 &-1 & 0 &-1 & -1\\ \end{pmatrix}$$

A universal Gröbner basis for the ideal associated to $\mathcal{L}^{-1}$ is given by

$$\sum_{i \in \mathrm{supp}(v)} v_i \prod_{j \in \mathrm{supp}(v) \smallsetminus \{ i \}} x_j,$$where $v=(v_1,\ldots,v_n)$ runs over the cocircuits of minimal support.

Label the columns of the matrix $v_1,\ldots,v_{10}$ and the rows $a,b,c,d,e$. The matroid generated by the dependence relations among $v_1,\ldots,v_{10}$ is the graphic matroid of $K_5$, so the minimal dependence relations among $v_1,\ldots,v_{10}$ correspond to the minimal sets of edges in $K_5$ containing a cycle, i.e. triangles. By definition, a cocircuit is a circuit in the dual matroid. Here, we want vectors of minimal support that will be killed by the circuits.

There are $\binom{5}{3}=10$ circuits. The circuits, together with the dependencies among the $v_i$ they correspond to, as well as the corresponding cocircuit are:

1,2,5 | $v_1+v_5-v_2$ | (1,-1,0,0,1,0,0,0,0,0) |

1,3,6 | $v_1+v_6-v_3$ | (1,0,-1,0,0,1,0,0,0,0) |

1,4,7 | $v_1+v_7-v_4$ | (1,0,0,-1,0,0,1,0,0,0) |

2,3,8 | $v_2+v_8-v_3$ | (0,1,-1,0,0,0,0,1,0,0) |

2,4,9 | $v_2+v_9-v_4$ | (0,1,0,-1,0,0,0,0,1,0) |

3,4,10 | $v_3+v_{10}-v_4$ | (0,0,1,-1,0,0,0,0,0,1) |

5,6,8 | $v_5+v_8-v_6$ | (0,0,0,0,1,-1,0,1,0,0) |

5,7,9 | $v_5+v_9-v_7$ | (0,0,0,0,1,0,-1,0,1,0) |

8,9,10 | $v_8+v_{10}-v_9$ | (0,0,0,0,0,0,0,1,-1,1) |

6,7,10 | $v_6+v_{10}-v_7$ | (0,0,0,0,0,1,-1,0,0,1) |

Then the universal Gröbner basis is $x_2x_5+x_1x_2-x_1x_5,x_3x_6+x_1x_3-x_1x_6,x_4x_7+x_1x_4-x_1x_7,x_3x_8+x_2x_3-x_2x_8,x_4x_9+x_2x_4-x_2x_9,x_4x_{10}+x_3x_4-x_3x_{10},x_6x_8+x_5x_6-x_5x_8,x_7x_9+x_5x_7-x_5x_9,x_9x_10+x_8x_9-x_8x_{10},x_7x_{10}+x_6x_7-x_6x_{10}$.