Veronese Varieties Research Articles

HYPERSURFACES IN 𝕊4THAT ARE OF Lk-2-TYPE

In this paper we begin the study of <TEX>$L_k$</TEX>-2-type hypersurfaces of a hypersphere <TEX>${\mathbb{S}}^{n+1}{\subset}{\mathbb{R}}^{n+2}$</TEX> for <TEX>$k{\geq}1$</TEX> Let <TEX>${\psi}:M^3{\rightarrow}{\mathbb{S}}^4$</TEX> be an orientable <TEX>$H_k$</TEX>-hypersurface, which is not an open portion of a hypersphere. Then <TEX>$M^3$</TEX> is of <TEX>$L_k$</TEX>-2-type if and only if <TEX>$M^3$</TEX> is a Clifford tori <TEX>${\mathbb{S}}^1(r_1){\times}{\mathbb{S}}^2(r_2)$</TEX>, <TEX>$r^2_1+r^2_2=1$</TEX>, for appropriate radii, or a tube <TEX>$T^r(V^2)$</TEX> of appropriate constant radius r around the Veronese embedding of the real projective plane <TEX>${\mathbb{R}}P^2({\sqrt{3}})$</TEX>.

Real rank with respect to varieties

We study the real rank of points with respect to a real variety X. This is a generalization of various tensor ranks, where X is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of X only. We show constructively that there exist varieties X for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman–Teitler on the maximal X-rank is tight. We also give examples of varieties X for which the gap between maximal complex and the maximal real rank is arbitrarily large. To facilitate our constructions we prove a conjecture of Reznick on the maximal real symmetric rank of symmetric bivariate tensors. Finally we study the geometry of the set of points of maximal real rank in the case of real plane curves.

Ideals of bounded rank symmetric tensors are generated in bounded degree

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.

Characters of equivariant 𝒟-modules on Veronese cones

Ford&gt;1d&gt;1, we consider the Veronese map of degreeddon a complex vector spaceWW,Verd:W⟶SymdW\mathrm {Ver}_d:W\longrightarrow \mathrm {Sym}^d W,w↦wdw\mapsto w^d, and denote its image byZZ. We describe the characters of the simpleGL(W)\mathrm {GL}(W)-equivariant holonomicD\mathcal {D}-modules supported on ZZ. In the case whend=2d=2, we obtain a counterexample to a conjecture of Levasseur by exhibiting aGL(W)\mathrm {GL}(W)-equivariantD\mathcal {D}-module on the Capelli type representationSym2W\mathrm {Sym}^2 Wwhich contains noSL(W)\mathrm {SL}(W)-invariant sections. We also study the local cohomology modulesHZ∙(S)H^{\bullet }_Z(S), whereSSis the ring of polynomial functions on the vector spaceSymdW\mathrm {Sym}^d W. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree∙=codim(Z)\bullet =\textrm {codim}(Z)), and moreover we prove that it is a simpleD\mathcal {D}-module and determine its character.

Representation stability for syzygies of line bundles on Segre–Veronese varieties

The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre–Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld’s conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.

Uniqueness of Nonnegative Tensor Approximations

We show that for a nonnegative tensor, a best nonnegative rank-r approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonnegative rank-one approximation, and that the set of nonnegative tensors with non-unique best rank-one approximations form an algebraic hypersurface. We show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor which does not satisfy this equation is guaranteed to have a unique best rank-one approximation. We also establish an analogue for real or nonnegative symmetric tensors. In addition, we prove a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can never be obtained by deflation. As an aside, we verify that the Euclidean distance (ED) discriminants of the Segre variety and the Veronese variety are hypersurfaces and give defining equations of these ED discriminants.

Connected Graphs Arising from Products of Veronese Varieties

Given a Segre squarefree Veronese configuration, following Bernd Sturmfels, we improve the study of the graphs associated to the configuration. We determine two special families of toric ideals and a finite set of moves for each of them, which still guarantee simultaneously the connection of all graphs arising from each family of moves.

Generalized varieties of sums of powers

Let $X\subset\mathbb{P}^{N}$ be an irreducible, non-degenerate variety. The generalized variety of sums of powers $VSP_H^X(h)$ of $X$ is the closure in the Hilbert scheme $Hilb_{h}(X)$ of the locus parametrizing collections of points $\{x_{1},...,x_{h}\}$ such that the $(h-1)$-plane $\left\langle x_{1},...,x_{h}\right\rangle$ passes trough a fixed general point $p\in\mathbb{P}^{N}$. When $X = V_{d}^{n}$ is a Veronese variety we recover the classical variety of sums of powers $VSP(F,h)$ parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of $VSP_H^X(h)$. In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of $VSP_H^X(h)$ are inherited from the birational geometry of variety $X$ itself.

Generalized varieties of sums of powers

Let X ⊂ PN be an irreducible, non-degenerate variety. The generalized variety of sums of powers V SPHX (h) of X is the closure in the Hilbert scheme Hilbh (X) of the locus parametrizing collections of points {x1,..., xh} such that the (h −1)-plane 〈x1,..., xh〉 passes through a fixed general point p ∈ PN. When X = Vdn n d is a Veronese variety we recover the classical variety of sums of powers V SP(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V SPHX (h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of VSPHX (h) are inherited from the birational geometry of variety X itself.

Higher distances for constant dimensions codes: the case of osculating spaces to a Veronese variety

In linear network coding the information is encoded in terms of a basis of a vector space and it is received as a basis of a possible altered vector space. In the constant dimension case Koetter and Kschischang introduced a metric on the Grassmannian and proved efficient and correct decoding in terms of this metric. Here we introduce a second order invariant of the code: the minimum dimension of the linear span of 3 different linear subspaces belonging to the code. This is the case \(s=3\) of a family \(d_s\) and \(d'_s\), \(s\ge 3\), of invariants of network codes. We study these invariants in a case recently proposed by Hansen (the set of all osculating spaces of a Veronese embedding of a finite projective space) and for a related case (the set of osculating spaces to curves of positive genus) with a complete description of the case of elliptic curves and the ones related to the Hermitian curve.

Veronese transform and Castelnuovo--Mumford regularity of modules

Veronese rings, Segre embeddings, or more generally Segre{Veronese embeddings are very important rings in algebraic geometry. In this paper we present an original, elementary way to compute the Hilbert{Poincar e series of these rings; as a consequence we compute their Castelnuovo{Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo{Mumford regularity of a Cohen{Macaulay nitely generated graded module,

Differential Forms on Quasihomogeneous Noncomplete Intersections

In this article, we discuss a few simple methods for computing the Poincare series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.

Bayesian Low-Rank and Sparse Nonlinear Representation for Manifold Clustering

Linear representation usually used by the optimization model about low-rankness and sparsity limits their applications to some extent. In this paper, we propose Bayesian low-rank and sparse nonlinear representation (BLSN) model exploiting nonlinear representation. Different from the optimization model, BLSN can be solved by traditional algorithm in Bayesian statistics easily without knowing the explicit mapping by kernel trick. Moreover, it can learn the parameters adaptively to choose the low-rank and sparse properties and also provides a way to enforce more properties on one quantity in a Bayesian model. Based on the observation that the data points drawn from a union of manifolds may gain more meaningful linear structure after a nonlinear mapping, we apply BLSN for manifold clustering. It can handle different problems by constructing various kernels. With respect to the case of linear manifold, known as subspace segmentation, we propose a kernel by the Veronese mapping. In addition, we also design the kernel matrices for the case of nonlinear manifold. Experimental results confirm the effectiveness and the potential of our model for manifold clustering.

On the Hilbert scheme of degeneracy loci of twisted differential forms

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.

Veronesean representations of Moufang planes

In 1901 Severi [18] proved that the complex quadric Veronese variety is determined by three algebraic/differential geometric properties. In 1984 Mazzocca and Me lone [10] obtained a combinatorial analogue of this result for finite quadric Veronese varieties. We make further abstraction of these properties to characterize Veronesean representations of all the Moufang projective planes defined over a quadratic alternative division algebra over an arbitrary field. In the process, new Veroneseans over a nonperfect field of characteristic 2 (related to purely inseparable field extensions) are found, and their corresponding projective representations of the associated groups studied. We show that these representations are indecomposable, but reducible, and determine their (irreducible) quotient and kernel.

Yet again on two examples by Iyama and Yoshino

We give an elementary proof of Iyama–Yoshino's classification of rigid maximal Cohen-Macaulay modules on Veronese embeddings in P 9 .

Non-linear maximum rank distance codes

By exploring some geometry of Segre varieties and Veronese varieties, new families of non-linear maximum rank distance codes and optimal constant rank codes are provided.

Horrocks correspondence on arithmetically Cohen–Macaulay varieties

We describe a vector bundle $\sE$ on a smooth $n$-dimensional ACM variety in terms of its cohomological invariants $H^i_*(\sE)$, $1\leq i \leq n-1$, and certain graded modules of "socle elements" built from $\sE$. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems where we construct vector bundles from these invariants and uniqueness theorems where we show that these data determine a bundle up to isomorphisms. The cases of the quadric hypersurface in $\mathbb P^{n+1}$ and the Veronese surface in $\mathbb P^5$ are considered in more detail.

Foulkes characters for complex reflection groups

Foulkes discovered a marvelous set of characters for the symmetric group by summing Specht modules of certain ribbon shapes according to height. These characters have many remarkable properties and have been the subject of many investigations, including a recent one [2] by Diaconis and Fulman, which established some new formulas, a conjecture of Isaacs, and a connection with Eulerian idempotents. We widen our consideration to complex reflection groups and find ourselves equipped from the start with a simple formula for (generalized) Foulkes characters which explains and extends these properties. In particular, it gives a factorization of the Foulkes character table which explains Diaconis and Fulman’s formula for the determinant, their link to Eulerian idempotents, and their formula for the inverse. We present a natural extension of a conjecture of Isaacs, and then use properties of Foulkes characters which resemble those of supercharacters to establish the result. We also discover a remarkable refinement of Diaconis and Fulman’s determinantal formula by considering Smith normal forms. Classic type A Foulkes characters have connections with adding random numbers, shuffling cards, the Veronese embedding, and combinatorial Hopf algebras [2, 7]. Our formula brings Orlik–Solomon coexponents from [12] the cohomology theory of [10] complements into the picture with the geometry of the Milnor fiber complex [8], and it gives rise to a curious classification at the end of the paper. The paper is structured as follows. Section 1 introduces Foulkes characters for Shephard and Coxeter groups. Key properties are quickly gathered, including our main formula. In Section 2, properties of type A Foulkes characters are explained and extended from the symmetric group to the infinite family of wreath products. In Section 3, Isaacs’ type A conjecture is sharpened for the Coxeter–Shephard–Koster family. Diaconis and Fulman’s type A determinantal formula is also extended here. Lastly, we determine exactly when the Foulkes characters are a basis for the space of class functions .g/ that depend only on the dimension of the fixed space of g.

On non-autonomous dynamical systems

In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the Poisson-Lie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.