Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S^2 V^* \otimes L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N-r, then the locus of points where the rank of the quadratic form is at most r-1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix identifies a Gysin map with the natural map from ordinary to relative cohomology.
Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and "nontwisted" Brill--Noether loci. We describe the tangent cones to the twisted Bril…
Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\bns $ is non-empty if $d= nd' +d'' $ with $0< d'' <2n$, $1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$. These results hold for an arbitrary curve of genus $\ge 2$, and allow us to construct a region in the associated ``Brill-Noether $\pa$-map'' of points for which the Brill-Noether loci are non-empty. Even for the generic case, the region so constructed extends beyond that defined by the so-called ``Teixidor parallelograms.'' For hyperelliptic curves, the same methods g…
Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a skew-symmetric bilinear form with values in a line bundle L and that Λ^2 V^* \otimes L is ample. Suppose that the maximum rank of the form at any point of X is r, where r>0 is even. The main result of this paper is that if d>2(N-r), then the locus of points where the rank of the form is at most r-2 is nonempty. This is a skew-symmetric analogue of the main result of math.AG/0305159; the proof is similar. If the hypothesis of ampleness is relaxed, we obtain a weaker estimate on the m…
We prove Lefschetz type theorems for cohomology groups and Picard groups of degeneracy loci for vector bundle maps. We also treat the case of antisymmetric maps.
We give two formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric degeneracy loci. We apply them in enumerative geometry, explore their algebraic combinatorics, and discuss K theory generalizations.
In [BFMT17] we introduced orbital degeneracy loci as generalizations of degeneracy loci of morphisms between vector bundles. Orbital degeneracy loci can be constructed from any stable subvariety of a representation of an algebraic group. In this paper we show that their canonical bundles can be conveniently controlled in the case where the affine coordinate ring of the subvariety is Gorenstein. We then study in a systematic way the subvarieties obtained as orbit closures in representations with finitely many orbits, and we determine the canonical bundles of the corresponding orbital degeneracy…
Corrado Segre played a leading role in the foundation of line geometry. We survey some recent results on degeneracy loci of morphisms of vector bundles where he still is of profound inspiration.
We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad'…
Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.
We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of Omega(2), the twisted cotangent bundle on P^(n-1), is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n=4.
We prove that, for 3 < m < n-1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Omega(2), the twisted cotangent bundle on P^{n-1}. For 3=m<n-1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parametrized by the Pfaffians of linear skew-symmetric matrices of order n.
Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility …
We generalize the prior linked symplectic Grassmannian construction, applying it to to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. We apply this general machinery to prove new results on nonemptiness and dimension of rank-2 Brill-Noether loci in a range of degrees.
The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smo…
Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbu…
Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.
We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.
The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strik…
We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these loci are given by explicit Pfaffian formulas. The Grassmannian formulas of Kazarian are important special cases, and the corresponding double Schubert polynomials of Ikeda, Mihalcea, and Naruse are shown to be equal to these Pfaffians.
We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types B, C, D which naturally extends the family of Lascoux of Schutzenberger in type A. These polynomials satisfy positivity, orthogonality, and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When w is a maximal Grassmannian element of the Weyl group, P_w(X,Y) can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type A formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows that there are no Chern class…
We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms.
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.
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…
This paper originated as an attempt to answer a question: what are the natural derived structures on Poisson degeneracy loci? We argue that the question could be possibly answered via a construction of differential graded operads that ``naturally'' act on the degeneracy loci. For each $m \ge 0$, we suggest what looks like a reasonable condition for a Poisson structure on a commutative differential graded algebra to be $m$-degenerate, i.e. to ``have rank $\le 2m$''. That condition will turn out to be a universal property of the operad that controls such Poisson algebras; we denote that operad $…
Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of rank $s-p$ over $W:=\{x\in V:\mathrm{rk} F(x)=p\}$. We associate with $(V,F)$ a descending chain of degeneracy loci of E (the generic polar varieties of $V$ represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded error probabilistic pseudo-polynomial time algor…
We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
Let V be a vector bundle on a scheme X endowed with a nondegenerate symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing maximal isotropic subbundles of V. The main goal of the paper is to give formulas for the classes of the loci in G parametrizing those subbundles which intersect successive members of of a fixed flag of isotropic subbundles of V in dimensions greater than 1,2,3,... . The formulas which we give are quadratic expressions in Q-polynomials of the tautological bundle on G and flag S-polynomials of the members of the flag. These formulas globalize to Lagrang…
In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from all isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian id…
We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank 2 vector bundle.
In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical line bundle of a Poisson manifold and its degeneracy loci---where the rank of the Poisson structure drops. As an application, we provide new evidence in favour of Bondal's conjecture that the rank \leq 2k locus of a Fano Poisson manifold always has dimension \geq 2k+1. In particu…
Given a generic map between flagged vector bundles on a Cohen-Macaulay variety, we construct maximal Cohen-Macaulay modules with linear resolutions supported on the Schubert-type degeneracy loci. The linear resolution is provided by the Schubert complex, which is the main tool introduced and studied in this paper. These complexes extend the Schubert functors of Kraśkiewicz and Pragacz, and were motivated by the fact that Schur complexes resolve maximal Cohen-Macaulay modules supported on determinantal varieties. The resulting formula in K-theory provides a "linear approximation" of the structu…
Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov operators. This proves and extends a conjecture of Gholamp…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formu…
We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon--Kempf--Laksov's determinantal formula and the latter generalize Pragacz--Kazarian's formula for the Chow ring. As an application, we introduce the factorial $GΘ/ GΘ'$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch--Kresch--Tam…
We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over …
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic descri…
In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elu…
Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We use this result to solve the constant rank problem for symmetric matrices, proving that the maximal dimension of a linear subspace of the space of $m\times m$ symmetric matrices such that each nonzero element has even rank $r \ge 2$ is $m-r+1$. We explain how this result relates t…
Degeneracy loci polynomials for quiver representations generalize several important polynomials in algebraic combinatorics. In this paper we give a nonconventional generating sequence description of these polynomials, when the quiver is of Dynkin type.
In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles.
We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety a…
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains a…
By analyzing degeneracy loci over projectivized vector bundles, we recompute the degree of the discriminant locus of a vector bundle and provide a new proof of the Bogomolov instability theorem.
We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.
