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A Gröbner basis for the graph of the reciprocal plane. (English) Zbl 1458.13022

An important invariant associated to any matroid and via this to any hyperplane arrangement is the charactersitic polynomial \[\chi_M(q)=\sum_{F\subseteq M}\mu(\hat{0},F)q^{r+1-r(F)},\] where the sum is over the flats of the matroid \(M\), \(\mu\) denotes the Möbius function, and \(r(F)\) is the rank of the flat \(F\).
H. Terao [J. Algebra 250, No. 2, 549–558 (2002; Zbl 1049.13011)] showed that in characteristic zero, the Hilbert series of the projective coordinate ring of the reciprocal plane \(R_\mathcal{A}\) of the hyperplane arrangement \(\mathcal{A}\) is given by \[H(R_\mathcal{A};t)=\sum_{i=0}^{r+1}w_i(t/1-t)^i,\] where \((-1)^iw_i\) is the coefficient of \(q^{r+1-i}\) in \(\chi_M(q)\).
On the other hand, J. Huh and E. Katz showed in [Math. Ann. 354, No. 3, 1103–1116 (2012; Zbl 1258.05021)] that in \(H^{2(2n-r)}(\mathbb{P}\times\mathbb{P})\) the cohomology class of \(\Gamma_{\mathcal{A}}\), the reciprocal graph of \(\mathcal{A}\) is given by \[[\Gamma_{\mathcal{A}}]=\sum_{i=0}^r\overline{w}_i [\mathbb{P}^{r-i}\times\mathbb{P}^i],\] where \((-1)^i\overline{w}_i\) is the coefficient of \(q^{r-i}\) in the reduced characteristic polynomial \[\overline{\chi}_M(q):=\chi_M(q)/(q-1).\]
The authors define in the present paper an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomial extracted from the Hilbert series in these two seemingly different manifestations of the characteristic polynomial agree.

MSC:

13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05E45 Combinatorial aspects of simplicial complexes
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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References:

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