Chow Form Research Articles

Chow forms and complete intersections in the projective space

We prove that every Hodge cohomology class of bidegree (q,q) on a projective manifold X can be recovered from its image by the Chow transformation restricted to a suitable irreducible algebraic component of the space Cq−1(X) of effective algebraic cycles in X of dimension q−1. An application to the problem of the approximation by algebraic cycles is given. In the case of the cohomology class of an effective algebraic cycle, this injectivity at the cohomological level is a consequence of the inversion formula for the Chow transform of a conormal. When X=PN, the inversion formula for the conormal is extended to the case of the conormal of any closed positive (q,q)-current. An inversion formula for the Radon transform, defined on the Grassmannian, of smooth functions is involved and is also used to obtain a characterization of Chow forms of complete intersections in the projective space, expressed by means of the Capelli differential operators.

On the Complexity of Chow and Hurwitz Forms

We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

On the construction of weakly Ulrich bundles

A well-known conjecture of Eisenbud, Schreyer and Weyman suggests that any projective variety carries an Ulrich (and hence also weakly Ulrich) bundle. This is known only in a handful of cases. In this paper I provide weakly Ulrich bundles on a class of surfaces and threefolds. I construct a weakly Ulrich bundle of large rank on any smooth complete surface in P3 over fields of characteristic p>0 and also for some classes of surfaces of general type in Pn. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in P4 of degree at most four, (2) more generally, Frobenius split Fano varieties of dimension at most three, (3) Frobenius split, smooth quintics in P4 (4) more generally, Frobenius split Calabi-Yau varieties of dimension at most three (5) Frobenius split (i.e. ordinary) abelian varieties of dimension at most three. These results also imply that Chow form of these varieties is the support of a single intrinsic determinantal equation.

Isotropic and coisotropic subvarieties of Grassmannians

We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead of conormal spaces.

Partial differential Chow forms and a type of partial differential Chow varieties

We first present an intersection theory of irreducible partial differential varieties with quasi-generic differential hypersurfaces. Then, we define partial differential Chow forms for irreducible partial differential varieties whose Kolchin polynomials are of the form And we establish for partial differential Chow forms most of the basic properties of their ordinary differential counterparts. Furthermore, we prove that a certain type of partial differential Chow varieties exist.Communicated by Jason Bell

Coisotropic hypersurfaces in Grassmannians

To every projective variety X, we associate a list of hypersurfaces in different Grassmannians, called the coisotropic hypersurfaces of X. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on tangent spaces. We present a new and simplified proof of that result. We show that the coisotropic hypersurfaces of X equal those of its projectively dual variety, and that their degrees are the polar degrees of X. Coisotropic hypersurfaces of Segre varieties are defined by hyperdeterminants, and all hyperdeterminants arise in that manner. We generalize Cayley's differential characterization of coisotropy and derive new equations for the Cayley variety which parametrizes all coisotropic hypersurfaces of given degree in a fixed Grassmannian. We provide a Macaulay2 package for transitioning between X and its coisotropic hypersurfaces.

The Chow Form of a Reciprocal Linear Space

A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.

Elimination Theory in Differential and Difference Algebra

Elimination theory is central in differential and difference algebra. The Wu-Ritt characteristic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems, the theory of differential Chow forms, and the theory of sparse differential and difference resultants.

On the bit complexity of polynomial system solving

We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.

Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension (p,p) on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents that are completely determined. Each of these currents is ddc-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah–Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.

The Chow form of the essential variety in computer vision

The Chow form of the essential variety in computer vision is calculated. Our derivation uses secant varieties, Ulrich sheaves and representation theory. Numerical experiments show that our formula can detect noisy point correspondences

Differential Chow varieties exist

The Chow variety is a parameter space for effective algebraic cycles on P n (or A n ) of given dimension and degree. We construct its analog for differential algebraic cycles on A n , answering a question of Gao, Li and Yuan [Trans. Amer. Math. Soc. 365 (2013) 4575–4632]. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties and algebraic varieties, the theory of prolongation spaces and the theory of differential Chow forms. In the course of the proof, several definability results from the theory of algebraically closed fields are required. Elementary proofs of these results are given in the appendix.

The Hurwitz form of a projective variety

The Hurwitz form of a variety is the discriminant that characterizes linear spaces of complementary dimension which intersect the variety in fewer than degree many points. We study computational aspects of the Hurwitz form, relate this to the dual variety and Chow form, and show why reduced degenerations are special on the Hurwitz polytope.

Facets of secondary polytopes and chow stability of toric varieties

Chow stability is one notion of Mumford’s Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope [7]. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is H-semistable if and only if it is H-polystable with respect to the standard complex torus action H . This essentially means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.

Theories and algorithms for difference and differential Chow form

Theories and algorithms for difference and differential Chow form

Ulrich bundles on rational surfaces with an anticanonical pencil

Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which means it has the same cone of cohomology table as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundle on rational surfaces with an anticanonical pencil, under a mild Brill-Noether assumption by using Lazarsfeld-Mukai bundles. Also we see that each of those surfaces carries its Chow form given by the Pfaffian of skew-symmetric morphism coming from an Ulrich bundle.

Computation of differential Chow forms for ordinary prime differential ideals

In this paper, we propose algorithms for computing differential Chow forms for ordinary prime differential ideals which are given by characteristic sets. The algorithms are based on an optimal bound for the order of a prime differential ideal in terms of a characteristic set under an arbitrary ranking, which shows the Jacobi bound conjecture holds in this case. Apart from the order bound, we also give a degree bound for the differential Chow form. In addition, for a prime differential ideal given by a characteristic set under an orderly ranking, a much simpler algorithm is given to compute its differential Chow form. The computational complexity of the algorithms is single exponential in terms of the Jacobi number, the maximal degree of the differential polynomials in a characteristic set, and the number of variables.

Macrophage depletion attenuates skin and kidney disease in lupus mice (BA7P.143)

Abstract Systemic lupus erythematosus (SLE) is an autoimmune disease characterized by the production of anti-nuclear autoantibodies and inflammatory damage in target organs, commonly the kidney and skin. Macrophages are thought to be important in the pathogenesis of lupus end organ disease, although studies to date have not been sufficiently conclusive. In this study, we investigated the role of macrophages in a spontaneous mouse model of SLE, the MRL/lpr mouse. PLX3397 (Plexxikon), a small molecule inhibitor of CSF1R, c-Kit, and FMS-like tyrosine kinase 3a, was administered in chow form to MRL/lpr mice starting at 9 weeks of age. Compared to matched control treated mice, PLX3397-treated MRL/lpr mice exhibited significantly less proteinuria and improved renal function. Furthermore, anti-dsDNA antibodies were significantly reduced at several time points. IBA-1 staining confirmed macrophage depletion within the kidneys of the PLX3397 treated mice. Interestingly, PLX3397-treated mice were also protected from the development of spontaneous skin lesions common to MRL/lpr mice, with control treated mice displaying significantly worse cutaneous disease macroscopically at 16 and 22 weeks of age. The improvement in skin lesions in PLX3397-treated mice was confirmed histologically in skin obtained at sacrifice. Our results indicate an important role for macrophages in the pathogenesis of skin and kidney disease in MRL/lpr mice, and implicate macrophages as a potential therapeutic target in SLE.

Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

Abstract The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that K ⁢ 3 K3 surfaces admit Ulrich bundles of every rank. We apply this to describe a pfaffian equation for the Chow form of a K ⁢ 3 K3 surface.

Difference Chow form

In this paper, the generic intersection theory for difference varieties is presented. Precisely, the intersection of an irreducible difference variety of dimension d>0 and order h with a generic difference hypersurface of order s is shown to be an irreducible difference variety of dimension d−1 and order h+s. Based on the intersection theory, the difference Chow form for an irreducible difference variety is defined. Furthermore, it is shown that the difference Chow form of an irreducible difference variety V is transformally homogeneous and has the same order as V.