Suppose $X$ is a closed sub-scheme of $Y$ and $Y$ is a closed sub-scheme of $Z$ that formally locally has an analog of a tubular neighborhood in a sense that we define in the paper. In this setting, we prove a formula for calculating the Segre class of $X$ in $Y$ in terms of the Segre class of $X$ in $Z$ and the Chern class of the normal bundle of $Y$ in $Z$. Intuitively, this means that we can obtain the Segre class of $X$ in $Y$ by first calculating the Segre class of $X$ in $Z$, and then "cancelling out" the contribution of the embedding of $Y$ in $Z$. It is important to note that the tubular neighborhood condition may be verified formally locally. As an application, we obtain a generalization of the Riemann Kempf formula to arbitrary integral curves.
We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.
We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is t…
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 prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among…
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 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 ideal…
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.
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 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 first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(Σ,H),\,H\in {\rm Pic}\,Σ$.
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.
We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.
We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes of rational polytopes.
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 bund…
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 math.AG/0203122, in certain special (but representative) cases.
Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler cha…
In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle.
Given a flat, projective morphism $Y \to T$ from an equidimensional scheme to a nonsingular curve and a subscheme $Z$ of $Y$, we give conditions under which specialization of the Segre class $s(N_{Z}Y)$ of the normal cone of $Z$ in $Y$ implies flatness of the normal cone. We apply this result to study when the relative tangent star cone of a flat family is flat.
Nakada's colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson's formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we utilize cohomological properties of Segre-MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada's formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Summing over weighted paths of…
We derive a formula for the Milnor class of scheme-theoretic global complete intersections (with arbitrary singularities) in a smooth variety in terms of the Segre class of its singular scheme. In codimension one the formula recovers a formula of Aluffi for the Milnor class of a hypersurface.
We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all smooth projective surfaces and dimension 1 quotients for surfaces X with p_g>0. We also show that the generating series of virtual cobordism classes can be irrational. Given a K-theory class on X of rank r, we associate natural series of virtual Segre a…
Let $V$ be a closed subscheme of a projective space $\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $ V$. The algorithm can be implemented using either symbolic or numerical methods. The algorithm is based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Using this result and known formulas for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps we…
The Macaulay2 package CharacteristicClasses provides commands for the computation of the topological Euler characteristic, the degrees of the Chern classes and the degrees of the Segre classes of a closed subscheme of complex projective space. The computations can be done both symbolically and numerically, the latter using an interface to Bertini. We provide some background of the implementation and show how to use the package with the help of examples.
If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{θ_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{θ_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field $k$ with $char(k) \neq 2$ such that $-1 \in {k^{\times}}^{2}$. This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension $d \geq 4$ over algebraically closed fields of characteristic $\neq 2$. As a consequence, we prove that any projective module of rank $3$ with trivial Chern classes o…
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…
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 give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
The goal of this work is to provide a fiber integration formula on the Demailly tower, that avoids step-by-step elimination of horizontal cohomology classes, and that yields computational effectivity. A natural twist of the Demailly tower is introduced and a recursive formula for the total Segre class at k-th level is obtained. Then, by interpreting single Segre classes as coefficients, an iterated residue formula is derived.
We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber-Novikov algebra in the complex cobordism ring of a point.
We give a classification of {\texttt{e.a.b.}} semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all {\texttt{e.a.b.}} semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of {\texttt{e.a.b.}} semistar (or star) operations we defined can all be distinct. In partic…
Let $\mathrm{Hilb}_nS$ be the Hilbert scheme of $n$ points on a smooth projective surface $S$. To a class $α\in K^0(S)$ correspond a tautological vector bundle $α^{[n]}$ on $\mathrm{Hilb}_nS$ and line bundle $L_{(n)}\otimes E^{\otimes r}$ with $L=\det(α)$, $r=\mathrm{rk}(α)$. In this paper we give closed formulas for the generating functions for the Segre classes $\int_{\mathrm{Hilb}_nS} s(α^{[n]})$, and the Verlinde numbers $χ(\mathrm{Hilb}_nS,L_{(n)}\otimes E^{\otimes r})$, for any surface $S$ and any class $α\in K^0(S)$. In fact we determine a more general generating function for $K$-theore…
We present a probabilistic algorithm to test if a homogeneous polynomial ideal $I$ defining a scheme $X$ in $\mathbb{P}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{\rm red}=V(\sqrt{I})$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requ…
The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (no…
Let $X_Σ$ be a smooth complete toric variety defined by a fan $Σ$ and let $V=V(I)$ be a subscheme of $X_Σ$ defined by an ideal $I$ homogeneous with respect to the grading on the total coordinate ring of $X_Σ$. We show a new expression for the Segre class $s(V,X_Σ)$ in terms of the projective degrees of a rational map specified by the generators of $I$ when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where $X_Σ$ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of $X_Σ$ via a dehomogenizing ideal we als…
This article can be seen as a sequel to the first author's article ``Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane'', where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of ``universality'' of certain formulas about the cohomology of the Hilbert schemes by definin…
We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their tautological integrals. In the fourfold case, we first construct these classes under additional conditions. We use novel methods relying on the wall-crossing of Joyce. Our results include -the dependence of the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface, -relations to the previous results in the literature, which addressed the …
We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form (k-p+1,k-p+2,...,k+1) or the form (k+p,k+p-1,...,k), and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the k-Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in …
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, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to an embedding 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 classe…
