A blow-up construction and graph coloring

In this note we construct a non-singular algebraic variety VG encoding the incidence information of a simple graph G, by a sequence of blow-ups of a projective space along suitable linear subspaces. The aim is to translate into the geometry of VG the combinatorial information about G; we nd that this can be done with surprising ease and e ciency. For example, we prove that the chromatic polynomial of the graph|that is, the polynomial giving for each m > 0 the number of ways in which G can be colored using m colors, so that no two adjacent vertices are assigned the same color|is (up to a power of the variable) the intersection product of a xed class in A1VG with a polynomial S(t) in PicVG[t]: the class is de ned as the Poincar e dual of the pull-back of the hyperplane class, with respect to a natural basis of PicVG, and S(t) is also easily de ned as a combination of the exceptional divisors arising in the blow-up construction. In x1 we describe the construction for graphs and state the above result precisely (but with no proofs), as a sales pitch for the rest of the paper, which examines the construction more carefully and gives deeper|but necessarily more technical|results. In fact the right level of generality to perform our construction is that of ‘combinatorial geometries which are projectively coordinatizable over some eld’; for short (and a little improperly) we will refer to these as matroids. Our construction can be performed starting from any (loopless) matroid embedded in a projective space, and specializes to the one in x1 for the cycle matroid of a graph. We give this more general construction in x2: roughly, the variety of a matroid is obtained by blowing-up the ambient projective space along the ats of the matroid, in order of increasing dimension. We prove the above result concerning the chromatic polynomial of a graph by showing that the characteristic polynomial of a matroid equals the intersection product of a xed 1-class by a suitable polynomial S(t) in the Pic of its variety. A question that then arises naturally regards the positivity of S(m) for a given m and a given class of matroids: we determine a large class (including cycle matroids of graphs) for which a close relative S(m) of S(m) is generated by global sections for all positive m. To support the point that our construction may o er a new angle on the theory of characteristic polynomials of matroids, in x3 we give ‘geometric proofs’ of a