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.

Plane curves with small linear orbits I

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary plane curves. Linear orbits of smooth plane curves are studied in [A-F1].Comment: 34 pages, 4 figures, AmS-TeX 2.1, requires xy-pic and eps

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety $X$, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; `stringy' Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a `change-of-variable' formula holds. v2: References added, and overly optimistic claims concerning non log-terminal singularities expunged.Comment: 42 pages, LaTeX, 2 figure

Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes

We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.Comment: 23 pages, final version. Dedicated to Robert MacPherson on the occasion of his 60th birthda

Chern classes of birational varieties

A theorem of Batyrev's asserts that if two nonsingular varieties V,W are birational, and their canonical bundles agree after pull-back to a resolution of indeterminacies of a birational map between them, then the Betti numbers of V and W coincide. We prove that, in the same hypotheses, the total homology Chern classes of V and W are push-forwards of the same class in the Chow group of the resolution. For example, it follows that the push-forward of the total Chern class of a crepant resolution of a singular variety is independent of the resolution.Comment: 8 pages, final version, to appear in IMR

Weighted Chern-Mather classes and Milnor classes of hypersurfaces

We introduce a class extending the notion of Chern-Mather class to possibly nonreduced schemes, and use it to express the difference between Schwartz-MacPherson's Chern class and the class of the virtual tangent bundle of a singular hypersurface of a nonsingular variety. Applications include constraints on the possible singularities of a hypersurface and on contacts of nonsingular hypersurfaces, and multiplicity computations.Comment: 15 page