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Algebraic Geometry

71. denote the Mobius function of this poset. Then Il(n) = IlL(l,n). 3 The Poincare Polynomial In this section we define one of the most important combinatorial invariants of an arrangement, its Poincare polynomial, and study its properties. 48 Let A be an arrangement with intersection poset L and Mobius function 11. Let t be an indeterminate. Define the Poincare polynomial of A by n(A, t) = Il(X)( _q(X). 47 that n(A, t) has nonnegative coefficients. In some cases it is easy to compute the values of 11 directly in order to obtain the Poincare polynomial.

It is easy to see that the map is rank preserving and surjective. In general, it is not injective. 0 Oriented Matroids Next we consider the special case of a real arrangement A. Recall that C(A) is the set of chambers of A. Thus M(A) = UCEC(AjC. 18 Let A be a real arrangement. Let C(A) = U C(AX). XEL(A) View C(A) as a collection of subsets of V. An element P E C(A) is a face. The support IFI of a face P is its affine linear span. Each face is open in its support. Let F denote the closure of P in V.

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