A blow-up construction and graph coloring

Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G (giving at each m>0 the number of colorings of G with m colors, such that no adjacent vertices are assigned the same color) can be computed as an intersection product between certain classes on these varieties, and other information such as Crapo's invariant find a very natural geometric counterpart. The note presents this construction, and gives `geometric' proofs of a number of standard combinatorial results on the chromatic polynomial.Comment: 22 pages, amstex 2.

Predegree Polynomials of Plane Configurations in Projective Space

2000 Mathematics Subject Classification: 14N10, 14C17.We work over an algebraically closed field of characteristic zero. The group PGL(4) acts naturally on PN which parameterizes surfaces of a given degree in P3. The orbit of a surface under this action is the image of a rational map PGL(4) ⊂ P15→PN. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general Pj, j being the dimension of the orbit. We find the predegrees and other invariants for all surfaces supported on unions of planes. The information is encoded in the so-called predegree polynomials , which possess nice multiplicative properties allowing us to compute the predegree (polynomials) of various special plane configurations. The predegree has both combinatorial and geometric significance. The results obtained in this paper would be a necessary step in the solution of the problem of computing predegrees for all surfaces