Tangential Variety Research Articles

On the secant varieties of tangential varieties

Let X ⊂ P r be an integral and non-degenerate variety. Let σ a , b ( X ) ⊆ P r , ( a , b ) ∈ N 2 , be the join of a copies of X and b copies of the tangential variety of X . Using the classical Alexander-Hirschowitz theorem (case b = 0 ) and a recent paper by H. Abo and N. Vannieuwenhoven (case a = 0 ) we compute dim ⁡ σ a , b ( X ) in many cases when X is the d -Veronese embedding of P n . This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that dim ⁡ σ 0 , b ( X ) is the expected one when X = Y × P 1 has a suitable Segre-Veronese style embedding in P r . As a corollary we prove that if d i ≥ 3 , 1 ≤ i ≤ n , and ( d 1 + 1 ) ( d 2 + 1 ) ≥ 38 the tangential variety of ( P 1 ) n embedded by | O ( P 1 ) n ( d 1 , … , d n ) | is not defective and a similar statement for P n × P 1 . For an arbitrary X and an ample line bundle L on X we prove the existence of an integer k 0 such that for all t ≥ k 0 the tangential variety of X with respect to | L ⊗ t | is not defective.

Cohen–Macaulay and Gorenstein tangential varieties of the Segre–Veronese varieties

We classify the tangential varieties of the Segre–Veronese varieties which are Cohen–Macaulay or Gorenstein.

Waring, tangential and cactus decompositions

We revise the famous algorithm for symmetric tensor decomposition due to Brachat, Comon, Mourrain and Tsidgaridas. Afterwards, we generalize it in order to detect possibly different decompositions involving points on the tangential variety of a Veronese variety. Finally, we produce an algorithm for cactus rank and decomposition, which also detects the support of the minimal apolar scheme and its length at each component.

Koszul modules and Green’s conjecture

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

Tangential varieties of Segre–Veronese surfaces are never defective

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.

Ranks on the Boundaries of Secant Varieties

In many cases (e.g. for many Segre or Segre-Veronese embeddings of multiprojective spaces) we prove (in characteristic 0) that a hypersurface of the b-secant variety of has X-rank > b. We prove it proving that the X-rank of a general point of the join of b − 2 copies of $X$ and the tangential variety of X is > b.

A generic uniqueness result for an interpolation problem for the join of a tangential variety gamma(x) and several copies of the variety X in P^2

Let X ⊂ ℙr be an integral and non-degenerate variety non-singular in codimension 1. Let τ(X,2) =τ(X) ⊂ ℙr be the tangential variety of X. Set n := dim(X). For any integer b>2 let τ(X,b)$ be the join of τ(X) and b-2 copies of X. Mimicking the notion of weakly k-degenerate varieties introduced by Chiantini and Ciliberto we give a conditions on X assuring that a general q ∈ τ(X,b) is in the linear span of a unique scheme Z ⊂ X with Z = v ∪ {p1, ... , pb-2} with v connected of degree 2 and v ⊂ Xreg.

Most secant varieties of tangential varieties to Veronese varieties are nondefective

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the d d th Veronese embedding of the projective n n -space P n \mathbb {P}^n have the expected dimension, modulo a few well-known exceptions. It is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of Alexander and Hirschowitz in 1995. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., d = 3 d=3 , the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension n n of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of P 79 \mathbb {P}^{79} in P 88559 \mathbb {P}^{88559} .

Real rank two geometry

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

Symmetrization of principal minors and cycle-sums

We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector there is an matrix that has all principal minors equal to . We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to treat three cases: symmetric, skew-symmetric and general square matrices. We describe the matrices that have such symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums. We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials.

A uniqueness result on the decompositions of a bi-homogeneous polynomial

In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial p (i.e. a partially symmetric tensor of where are two complex, finite-dimensional vector spaces) if its rank with respect to the Segre–Veronese variety is at most . Such a polynomial may not have a unique minimal decomposition as with and coefficients, but we can show that there exist unique , , two unique linear forms , , and two unique bivariate polynomials and such that either or , ( being appropriate coefficients). In the second part of the paper we focus on the tangential variety of the Segre–Veronese varieties. We compute the rank of their tensors (that is valid also in the case of Segre–Veronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the two factors Segre–Veronese varieties.

Secants of minuscule and cominuscule minimal orbits

We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian.

Secant Cumulants and Toric Geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.

Tangential varieties of Segre–Veronese varieties

We determine the minimal generators of the ideal of the tangential variety of a Segre–Veronese variety, as well as the decomposition into irreducible $${GL}$$ -representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.

COMBINATORIAL DEGREE BOUND FOR TORIC IDEALS OF HYPERGRAPHS

Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform hypergraphs in terms of balanced hypergraph bicolorings, separators, and splitting sets. In turn, this provides complexity bounds for algebraic statistical models associated to hypergraphs. As two main applications, we recover a well-known complexity result for Markov bases of arbitrary 3-way tables, and we show that the defining ideal of the tangential variety is generated by quadratics and cubics in cumulant coordinates.

Products of Young symmetrizers and ideals in the generic tensor algebra

We describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau, generalizing the classical quasi-idempotence of Young symmetrizers. We derive some consequences to the structure of ideals in the generic tensor algebra and its partial symmetrizations. Instances of these generic algebras appear in the work of Sam and Snowden on twisted commutative algebras, as well as in the work of the author on the defining ideals of secant varieties of Segre---Veronese varieties, and in joint work of Oeding and the author on the defining ideals of tangential varieties of Segre---Veronese varieties.

Binary Cumulant Varieties

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2×2×···×2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary cumulants. Our primary focus lies on hidden subset models. Parametrizations and implicit equations in cumulants are derived for hyperdeterminants, for secant and tangential varieties of Segre varieties, and for certain context-specific independence models. Extending work of Rota and collaborators, we explore the polynomial inequalities satisfied by cumulants.

Tensor ranks on tangent developable of Segre varieties

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

On the X-rank with respect to linearly normal curves

In this paper we study the X-rank of points with respect to smooth linearly normal curves $${X \subset \mathbb {P}^{n}}$$ of genus g and degree n+g. We prove that, for such a curve X, under certain circumstances, the X-rank of a general point of X-border rank equal to s is less or equal than n + 1 − s. In the particular case of g = 2 we give a complete description of the X-rank if n = 3, 4; while if n ≥ 5 we study the X-rank of points belonging to the tangential variety of X.

Set-theoretic defining equations of the tangential variety of the Segre variety

We prove a set-theoretic version of the Landsberg–Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture, we use a connection to the author’s previous work and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than 1. We discuss applications to semiseparable matrices, tensor rank versus border rank, context-specific independence models and factor analysis models.