We review the basic definitions in Fulton-MacPherson Intersection Theory and discuss a theory of `characteristic classes' for arbitrary algebraic varieties, based on this intersection theory. We also discuss a class of graph invariants motivated by amplitude computations in quantum field theory. These `abstract Feynman rules' are obtained by studying suitable invariants of hypersurfaces defined by the Kirchhoff-Tutte-Symanzik polynomials of graphs. We review a `motivic' version of these abstract Feynman rules, and describe a counterpart obtained by intersection-theoretic techniques.
We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space $\mathcal M_{0,4}$. We also conjecture that the corresponding polynomials are {\em log-concave,\/} on the basis of hundreds of explicit computations.
Motivic Chern classes are elements in the K theory of an algebraic variety $X$ depending on an extra parameter $y$. They are determined by functoriality and a normalization property. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory flag manifolds $G/B$. The calculation is recursive starting from the class of a point, and using the Demazure-Lusztig operators in the Hecke algebra. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Feh{\'e}r, Rim{\'a}nyi and Weber. The Hecke algebra action on the K-theory of the dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the Poincar{\'e} dual version of the motivic Chern class to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations, and holomorphic properties, of the coefficients in the transition matrix between the standard and the Casselman's basis.
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex algebraic manifold X, the homogenized, torus equivariant CSM class of a constructible function φ is the restriction of the characteristic cycle of φ via the zero section of the cotangent bundle of X. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize X to be a (generalized) flag manifold G/B. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke orthogonality' of CSM classes, determined by the DL operator and its Poincar{\'e} adjoint. We further use the theory of holonomic D_{X}-modules to show that the characteristic cycle of the Verma module, restricted to the zero section, gives the CSM class of a Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds G/P.
Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on~$X$ with effective characteristic cycle is effective. This extends and unifies several previous results in the literature, and yields several new results. For example, we prove that Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety is effective; that the intersection homology Chern class of the theta divisor for a non-hyperellptic curve is signed-effective; and we prove more general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds. The latter result proves and generalizes a variant of a conjecture of Feh{\'e}r and Rim{\'a}nyi. Finally, we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.
Given a homogeneous ideal in a polynomial ring over C, we adapt the construction of Newton-Okounkov bodies to obtain a convex subset of Euclidean space such that a suitable integral over this set computes the Segre zeta function of the ideal. That is, we extract the numerical information of the Segre class of a subscheme of projective space from an associated (unbounded) Newton-Okounkov convex set. The result generalizes to arbitrary subschemes of projective space the numerical form of a previously known result for monomial schemes.
Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its singular locus and, in the hypersurface case, Milnor numbers. We obtain simple proofs of these formulas, recovering along the way several classically known results.
There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic 0) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years. We generalize this formula to arbitrary embeddable schemes: for every subscheme X of a nonsingular variety V, we define an associated subscheme Y of a projective bundle over V and provide an explicit formula for the Chern-Schwartz-MacPherson class of X in terms of the Segre class of Y. If X is a local complete intersection, a version of the result yields a direct expression for the Milnor class of X.
We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.
We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.
We define a power series associated with a homogeneous ideal in a
polynomial ring, encoding information on the Segre classes defined by
extensions of the ideal in projective spaces of arbitrarily high
dimension. We prove that this power series is rational, with poles
corresponding to generators of the ideal, and with numerator of
bounded degree and with nonnegative coefficients. We also prove that
this `Segre zeta function' only depends on the integral closure of the
ideal.
The results follow from good functoriality properties of the `shadows'
of rational equivalence classes of projective bundles. More precise
results can be given if all homogeneous generators have the same
degree, and for monomial ideals.
In certain cases, the general description of the Segre zeta function
given here leads to substantial improvements in the speed of
algorithms for the computation of Segre classes. We also compute the
projective ranks of a nonsingular variety in terms of the
corresponding zeta function, and we discuss the Segre zeta function of
a local complete intersection of low codimension in projective space.
We study a class obtained from the Segre class s(Z,Y) of an embedding of schemes by incorporating the datum of a line bundle on Z. This class satisfies basic properties analogous to the ordinary Segre class, but leads to remarkably simple formulas in standard intersection-theoretic situations such as excess or residual intersections. We prove a formula for the behavior of this class under linear joins, and use this formula to prove that a `Segre zeta function' associated with ideals generated by forms of the same degree is a rational function.
We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a `Segre-Bertini' theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections. These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals. The Segre-Bertini theorem has applications to characteristic classes of singular varieties. The main results are motivated by the problem of computing Segre classes explicitly and applications of Segre classes to enumerative geometry.
We obtain an algorithm describing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in generalized flag manifolds $G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class of a Schubert cell is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold $G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples.
We express the Segre class of a monomial scheme---or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections---in terms of an integral computed over an associated body in euclidean space. The formula is in the spirit of the classical Bernstein-Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of `r.c.-monomial schemes'.
We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes of related polytopes, in the spirit of the classical Bernstein-Kouchnirenko theorem, but extending the scope of these formulas to more general monomial maps. We also determine a condition under which the multidegrees may be computed in terms of the characteristic polynomial of an associated matrix.
We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a 'splayedness' assumption. The relation is shown to hold for both the Chern-Schwartz-MacPherson class and the Chern-Fulton class. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.
We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stem from applications to `matrix rigidity'; we also view them as an excellent source of examples to test methods in intersection theory, particularly computations of Segre classes. Our results are generally expressed in terms of intersection numbers in Grassmannians, which can be computed explicitly in many cases. We observe that, surprisingly (to us), these degrees appear to match the numbers of Kekul\'e structures of certain `benzenoid hydrocarbons', and arise in many other contexts with no apparent direct connection to the enumerative geometry of rank conditions.
We express the Segre class of a monomial scheme in projective space in terms of log canonical thresholds of associated ideals. Explicit instances of the relation amount to identities involving the classical polygamma functions.
We give a very informal presentation of background on the Grothendieck group of varieties and on characteristic classes, both viewed as generalizations of the ordinary topological Euler characteristic. We then review some recent work using these tools to study `graph hypersurfaces'---a topic motivated by the algebro-geometric interpretation of Feynman amplitudes as periods of complements of these hypersurfaces. These notes follow closely, both in content and style, my lectures at the Summer school in Villa de Leyva, July 5-8, 2011.
We obtain several new characterizations of splayedness for divisors: a Leibniz property for ideals of singularity subschemes, the vanishing of a `splayedness' module, and the requirements that certain natural morphisms of modules and sheaves of logarithmic derivations and logarithmic differentials be isomorphisms. We also consider the effect of splayedness on the Chern classes of sheaves of differential forms with logarithmic poles along splayed divisors, as well as on the Chern-Schwartz-MacPherson classes of the complements of these divisors. A postulated relation between these different notions of Chern class leads to a conjectural identity for Chern-Schwartz-MacPherson classes of splayed divisors and subvarieties, which we are able to verify in several template situations.
We propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables.
We obtain a precise relation between the Chern-Schwartz-MacPherson
class of a subvariety of projective space and the Euler
characteristics of its general linear sections. In the case of a
hypersurface, this leads to simple proofs of formulas of
Dimca-Papadima and Huh for the degrees of the polar map of a
homogeneous polynomial, extending these formula to any algebraically
closed field of characteristic~$0$, and proving a conjecture of
Dolgachev on `homaloidal' polynomials in the same context. We
generalize these formulas to subschemes of higher codimension in
projective space.
We also describe a simple approach to a theory of `polynomial
Chern classes' for varieties endowed with a morphism to projective
space, recovering properties analogous to the Deligne-Grothendieck
axioms from basic properties of the Euler characteristic. We prove
that the polynomial Chern class defines homomorphisms from suitable
relative Grothendieck rings of varieties to Zbb[t].
We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion-contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.
We study the behavior of the Chern classes of graph hypersurfaces
under the operation of deletion-contraction of an edge of the
corresponding graph. We obtain an explicit formula when the edge
satisfies two technical conditions, and prove that both these
conditions hold when the edge is multiple in the graph. This leads to
recursions for the Chern classes of graph hypersurfaces for graphs
obtained by adding parallel edges to a given (regular) edge.
Analogous results for the case of Grothendieck classes of graph
hypersurfaces were obtained in previous work. Both Grothendieck
classes and Chern classes were used to define `algebro-geometric'
Feynman rules. The results in this paper provide further evidence
that the polynomial Feynman rule defined in terms of the
Chern-Schwartz-MacPherson class of a graph hypersurface reflects
closely the combinatorics of the corresponding graph.
The key to the proof of the main result is a more general formula for the
Chern-Schwartz-MacPherson class of a transversal intersection
(see section 3), which may be of independent interest.
We also describe a more geometric approach, using the apparatus of
`Verdier specialization'.
The Chern class of the sheaf of logarithmic derivations along a simple normal crossing divisor equals the Chern-Schwartz-MacPherson class of the complement of the divisor. We extend this equality to more general divisors, which are locally analytically isomorphic to free 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.
Let X in V be a closed embedding, with V - X nonsingular. We define
a constructible function on X, agreeing with Verdier's specialization
of the constant function 1 when X is the zero-locus of a function on
V. Our definition is given in terms of an embedded resolution of X;
the independence on the choice of resolution is obtained as a
consequence of the weak factorization theorem of Abramovich et al.
The main property of the specialization function is a compatibility
with the specialization of the Chern class of the complement V-X. With
the definition adopted here, this is an easy consequence of standard
intersection theory. It recovers Verdier's result when X is the
zero-locus of a function on V.
Our definition has a straightforward counterpart in a motivic
group. The specialization function and the corresponding Chern class
and motivic aspect all have natural `monodromy' decompositions, for
for any X in V as above.
The definition also yields an expression for Kai Behrend's
constructible function when applied to (the singularity subscheme of)
the zero-locus of a function on V.
We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for graphs that are chains of polygons and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletion-contraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorifications of graph invariants.
We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes--Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules. We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.
The subring of the Grothendieck ring of varieties generated by the
graph hypersurfaces of quantum field theory maps to the monoid ring of
stable birational equivalence classes of varieties. We show that the
image of this map is the copy of Z generated by the class of a
point.
Thus, the span of the graph hypersurfaces in the Grothendieck ring
is nearly killed by setting the Lefschetz motive L to zero,
while it is known that graph hypersurfaces generate the Grothendieck
ring over a localization of Z[L] in which L
becomes invertible. In particular, this shows that the graph
hypersurfaces do
not generate the Grothendieck ring prior to localization.
The same result yields some information on the mixed Hodge
structures of graph hypersurfaces, in the form of a constraint on the
terms in their Deligne-Hodge polynomials.
The purpose of this paper is to show that, under certain combinatorial conditions on the graph, parametric Feynman integrals can be realized as periods on the complement of the determinant hypersurface in an affine space depending on the number of loops of the Feynman graph. The question of whether the Feynman integrals are periods of mixed Tate motives can then be reformulated (modulo divergences) as a question on a relative cohomology being a realization of a mixed Tate motive. This is the cohomology of the pair of the determinant hypersurface complement and a normal crossings divisor depending only on the number of loops and the genus of the graph. We show explicitly that this relative cohomology is a realization of a mixed Tate motive in the case of three loops and we give alternative formulations of the main question in the general case, by describing the locus of intersection of the divisor with the determinant hypersurface complement in terms of intersections of unions of Schubert cells in flag varieties. We also discuss different methods of regularization aimed at removing the divergences of the Feynman integral.
We present new explicit constructions of weak coupling limits of F-theory generalizing Sen's construction to elliptic fibrations which are not necessary given in a Weierstrass form. These new constructions allow for an elegant derivation of several brane configurations that do not occur within the original framework of Sen's limit, or which would require complicated geometric tuning or break supersymmetry. Our approach is streamlined by first deriving a simple geometric interpretation of Sen's weak coupling limit. This leads to a natural way of organizing all such limits in terms of transitions from semistable to unstable singular fibers. These constructions provide a new playground for model builders as they enlarge the number of supersymmetric configurations that can be constructed in F-theory. We present several explicit examples for E8, E7 and E6 elliptic fibrations.
We extend the classical formula of Porteous for blowing-up Chern classes
to the case of blow-ups of possibly singular varieties along regularly
embedded centers.
The proof of this generalization is perhaps conceptually simpler than the
standard argument for the nonsingular case, involving Riemann-Roch
without denominators. The new approach
relies on the explicit computation of an ideal, and a mild generalization
of a well-known formula for the normal bundle of a proper transform.
We also discuss alternative, very short proofs of the standard
formula in some cases: an approach relying on the theory of
Chern-Schwartz-MacPherson classes (working in characteristic 0), and
an argument reducing the formula to a straightforward computation of
Chern classes for sheaves of differential 1-forms with logarithmic
poles (when the center of the blow-up is a complete intersection).
Every complex plane curve C determines a subscheme S
of the P^{8} of 3x3 matrices, whose projective
normal cone (PNC) captures subtle invariants of C.
In a previous paper (FSU07-15) we obtain a set-theoretic
description of the PNC and thereby we determine all possible limits of
families of plane curves whose general element is isomorphic to
C. The main result of this article is the determination of the
PNC as a cycle; this is an essential ingredient in our
computation in Linear orbits of arbitrary plane curves,
Michigan Math J., 48 (2000) 1-37, of the degree of the
PGL(3)-orbit closure of an arbitrary plane curve, an invariant of
natural enumerative significance.
We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of a subscheme, determined by the curve, of the P^{8} of 3x3 matrices. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in Linear orbits of arbitrary plane curves, Michigan Math J., 48 (2000) 1-37.
We consider the infinite family of Feynman graphs known as the ``banana graphs'' and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern--Schwartz--MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.
In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are essential in both the physics and the mathematics arguments: the tadpole relation may be interpreted as an identity involving stringy invariants of a singular hypersurface, and corrections for the presence of pinch-points. The mathematical discussion is streamlined by the use of Chern-Schwartz-MacPherson classes of singular varieties. We also show how the main identity may be obtained by applying `Verdier specialization' to suitable constructible functions.
We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of $d$-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of $d$.
We give a short proof of the fact that the Chern classes for singular varieties defined by Marie-Hélène Schwartz by means of "radial frames" agree with the functorial notion defined by Robert MacPherson.
We introduce a formal integral on the system of varieties mapping
properly and birationally to a given one, with value in an associated
Chow group. Applications include comparisons of Chern numbers of
birational varieties, new birational invariants, `stringy' Chern
classes, and a `celestial' 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.
The formalism has points of contact with motivic integration.
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.
We introduce the notion of {\em proChow group\/} of varieties, agreeing with the notion of Chow group for complete varieties and covariantly functorial with respect to {\em arbitrary\/} morphisms. We construct a natural transformation from the functor of constructible functions to the proChow functor, extending MacPherson's natural transformation. We illustrate the result by providing very short proofs of (a generalization of) two well-known facts on Chern-Schwartz-MacPherson classes.
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.
We study different notions of blow-up of a scheme X along a
subscheme Y, depending on the datum of an embedding of X into an
ambient scheme. The two extremes in this theory are the ordinary
blow-up, Bl_{Y}X, corresponding to the identity X -> X,
and the `quasi-symmetric blow-up', Bl_{Y}X, corresponding to an
embedding X -> M into a nonsingular variety M. We prove that this latter
blow-up is intrinsic of Y and X, and is universal with respect to the
requirement of being embedded as a subscheme of the ordinary blow-up of
some ambient space along Y.
We consider these notions in the context of the theory of characteristic
classes of singular varieties. We prove that if X a hypersurface in a
nonsingular variety and Y is its `singularity subscheme', these two
extremes embody respectively the conormal and characteristic
cycles of X. Consequently, the first carries the essential information
computing Chern-Mather classes, and the second is likewise a carrier for
Chern-Schwartz-MacPherson classes. In our approach, these classes are
obtained from Segre class-like invariants, in precisely the same way as
other intrinsic characteristic classes such as those proposed by William
Fulton, and by W. Fulton and Kent Johnson.
We also identify a condition on the singularities of a hypersurface
under which the quasi-symmetric blow-up is simply the linear fiber space
associated with a coherent sheaf.
Let $φ: V\dashrightarrow W$ be a birational map between
smooth algebraic varieties which does not change the canonical class
(in the sense of Batyrev). We prove that the total homology Chern
classes of $V$ and $W$ are push-forwards of the same class from a
resolution of indeterminacies of $φ$.
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.
We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, nonreduced) projective scheme. For example, the algorithm yields the topological Euler characteristic of the support of a projective scheme $S$, given the homogeneous ideal of $S$. The algorithm has been implemented in Macaulay2, and it is available here.
Considerations based on the known relation between different characteristic classes for singular hypersufaces suggest that a form of the `inclusion-exclusion' principle may hold for Segre classes. We formulate and prove such a principle for a notion closely related to Segre classes. This is used to provide a simple computation of the classes introduced in our previous work, in certain special (but representative) cases.
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 an arbitrary scheme in terms of this class.
We show that the Chern-Schwartz-MacPherson class of a hypersurface X in a nonsingular variety M `interpolates' between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern-Schwartz-MacPherson class of such `nice' hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern-Schwartz-MacPherson class of X in terms of its polar classes.
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.
The `linear orbit' of a plane curve of degree d is its orbit in \P^{d(d+3)/2} under the natural action of \PGL(3). In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space \P^8 of 3x3 matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work.
The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of curves supported on unions of lines. Together with the results of [3], this encompasses the enumerative geometry of all plane curves with small linear orbit. This information will serve elsewhere as an ingredient in the computation of the degree of the orbit closure of an arbitrary plane curve.
The `linear orbit' of a plane curve of degree d is its
orbit in P^{d(d+3)/2} 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].
We express the Chern-Schwartz-MacPherson class of a possibly singular variety in terms of the total Chern class of a bundle of forms with logarithmic poles. As an application, we obtain a formula for the Chern-Schwartz-MacPherson class of a hypersurface of a nonsingular variety, in terms of the Chern-Mather class of a suitable sheaf.
We prove a formula relating Chern-Schwartz-MacPherson's class of a hypersurface in a nonsingular variety to other definitions of homology Chern classes of singular varieties, such as Mather's Chern class and a class introduced by W. Fulton.
We compute the Euler obstruction and Mather's Chern class of the discriminant hypersurface of a very ample linear system on a nonsingular variety. Comparing the codimension-1 and 2 terms of this and other characteristic classes of the discriminant leads to a quick computation of the degrees of the loci of cuspidal and binodal sections of a very ample line bundle on a smooth variety, and of the tacnodal locus for linear systems on a surface. We also compute explicitly all terms in the Schwartz-MacPherson's classes of strata of the discriminant of cubic plane curves, and of the discriminants of O(d) on P^1.
This ridiculously short note is devoted to the proof of the following
fact: if \alpha is a class of rank r in the Grothendieck group of
vector bundles over a scheme, and L is a line bundle, then
c_{r+1}(\alpha) = c_{r+1}(\alpha\otimes [L]).
The proof is elementary. Maybe the most interesting thing about this
is that it shows up with surprising frequence in intersection-theoretic
computations inspired by enumerative geometry.
Let Y be the singular locus of a hypersurface X in a smooth variety M,
with the scheme structure defined by the Jacobian ideal of X (we will
say then that Y is the singular {\it scheme\/} of X, to emphasize that
the scheme structure of Y is important for our considerations). In
this note we consider a class in the Chow group of~Y which arises
naturally in this setup, and which captures much
intersection-theoretic information about the situation. We produce
constraints for a given scheme to be a singular scheme of a
hypersurface, and we obtain applications to duality recovering, and
sometime strengthening, results of Holme, Landman, Ein, and Zak.
The degree of the dimension-0 component of our class agrees with a
known generalization, due to Parusinski, of the usual Milnor number.
Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of algebraic varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G can be computed as an intersection product of certain classes on these varieties, or recovered in terms of the Segre classes of related subschemes of a projective space; other information such as Crapo's invariant also finds 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 and Crapo's invariant.
We prove that MacPherson's total Chern class of a singular
hypersurface agrees `numerically' with a class obtained by means of
Fulton's intrinsic class of a scheme. More precisely, for a
hypersurface X with Jacobian scheme J, and any positive integer t,
consider the class P(X,J,t) obtained by taking Fulton's class of the
t-thickening of X along J. Then P(X,J,t) is a polynomial in t (with
coefficients in the Chow group of X), and we show that P(X,J,-1)
agrees with MacPherson's class of X after push-forward via the map
defined by the linear system of X.
We conjecture the equality holds at the level of Chow groups, and
speculate that a similar result should hold for arbitrary algebraic
schemes in characteristic 0.
Orbit closures of sets of points of the projective line under the action of the automorphism group of the latter are studied in terms of their degree and multiplicity along their boundary.
We compute the multiplicity of the discriminant of a line bundle L over a nonsingular variety S at a given section X, in terms of the Chern classes of L and of the cotangent bundle of S, and the Segre classes of the jacobian scheme of X in S. For S a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a aterm that depends on the Milnor numbers of the singularities of X_{red}. Also, under certain hypotheses, we provide fromulae for the `higher discriminants' that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be `small'.
A desingularization of the orbit closures of plane curves under the action of the automorphism group of the plane is constructed and used to study such orbits, obtaining the degree of the orbit closure of an arbitrary smooth curve in terms of its degree and of the nature of its flexes and its automorphism group. The result has a transparent enumerative interpretation.
We use a sequence of blow-ups over the projective space parametrizing plane curves of a degree d to obtain some enumerative results concerning smooth plane curves of arbitrary degree. For d=4, this gives a first modern verification of results of H. G. Zeuthen.
We employ a variety of complete cubics to give formulas for the characteristic numbers of families parametrized by hypersurfaces F in the P^{9} of plane cubics, in terms of infromation easily accessible given the equation of F. As examples, we obtain explicit results for families of cubics with given j-invariant and for other families arising naturally from the geometry of plane cubics.
Two blow-ups over the projective space P^^{N} parametrizing plane curves of a given degree yield a compactification of the space of reduced curve used elsewhere to obtain partial enumerative results for families of nonsingular plane curves. In this paper it is shown how to employ the construction to obtain enumerative results for families of plane curves with a node or a cusp. The results recover known results for cubics, give a first modern verification of some computation of Zeuthen's for quartics, and are new for higher degree. The heart of the computation is the derivation of key Segre classes relating the intersection calculus at the different stages of the blow-up construction.
The variety of complete cubics obtained elsewhere is used to recover classical enumerative results for singular plane cubics, and obtain new results for cubic curves with prescribed conditions with respects to flags in the plane.
We construct a variety of complete plane cubics by a sequence of five blow-ups over P^{9}. This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.
The characteristic numbers for the family of smooth plane cubics are computed, verifying results of Maillard and Zeuthen.
We survey some notions of characteristic classes for singular varieties, with particular attention devoted to the recently introduced notion of `stringy' Chern classes.