717 research outputs found
Grothendieck classes and Chern classes of hyperplane arrangements
We show that the characteristic polynomial of a hyperplane arrangement can be
recovered from the class in the Grothendieck group of varieties of the
complement of the arrangement. This gives a quick proof of a theorem of Orlik
and Solomon relating the characteristic polynomial with the ranks of the
cohomology of the complement of the arrangement.
We also show that the characteristic polynomial can be computed from the
total Chern class of the complement of the arrangement; this has also been
observed by Huh. In the case of free arrangements, we prove that this Chern
class agrees with the Chern class of the dual of a bundle of differential forms
with logarithmic poles along the hyperplanes in the arrangement; this follows
from work of Mustata and Schenck. We conjecture that this relation holds for
all free divisors.
We give an explicit relation between the characteristic polynomial of an
arrangement and the Segre class of its singularity (`Jacobian') subscheme. This
gives a variant of a recent result of Wakefield and Yoshinaga, and shows that
the Segre class of the singularity subscheme of an arrangement together with
the degree of the arrangement determine the ranks of the cohomology of its
complement.
We also discuss the positivity of the Chern classes of hyperplane
arrangements: we give a combinatorial interpretation of this phenomenon, and we
discuss the cases of generic and free arrangements.Comment: 21 pages, minor revision. To appear in IMR
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
Inclusion-exclusion and Segre classes
We propose a variation of the notion of Segre class, by forcing a naive
`inclusion-exclusion' principle to hold. The resulting class is computationally
tractable, and is closely related to Chern-Schwartz-MacPherson classes. We
deduce several general properties of the new class from this relation, and
obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page
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.
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
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 , 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
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
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