Secant Varieties Research Articles

Singularities of secant varieties from a Hodge theoretic perspective

AbstractWe study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have ‐Du Bois singularities (in a sense that was proposed in Shen, Venkatesh, and Vo [On k‐Du Bois and k‐rational singularities, arXiv e‐prints (June 2023), arXiv:2306.03977]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre‐1‐rational. From these results, we deduce several consequences, including a Kodaira–Akizuki–Nakano type vanishing result for the reflexive differential forms of the secant varieties.

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.

Syzygies of secant varieties of curves of genus 2

Ein, Niu and Park showed in [6] that if the degree of the line bundle L on a curve of genus g is at least 2g+2k+1, the k-th secant variety of the curve via the embedding defined by the complete linear system of L is normal, projectively normal and arithmetically Cohen-Macaulay, and they also proved some vanishing of the Betti diagrams. However, the length of the linear strand of weight k+1 of the resolution of the secant variety Σk of a curve of g≥2 is still mysterious. In this paper we calculate the complete Betti diagrams of the secant varieties of curves of genus 2 using Boij-Söderberg theory. The main idea is to find the pure diagrams that contribute to the Betti diagram of the secant variety via calculating some special positions of the Betti diagram.

Joins, Secant Varieties and Their Associated Grassmannians

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins are defined by taking a closure. We study the set obtained before making the closure (often called the open part of the join) and the set added after making the closure (called the boundary of the join). For a point q of the open part, we give conditions for the uniqueness of the set proving that q is in the open part.

Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

Identifiability and singular locus of secant varieties to Grassmannians

Identifiability and singular locus of secant varieties to Grassmannians

A Note on Secant-Defective Varieties and Clifford Modules

We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo’s Divisibility Property for LQEL varieties whose secant varieties do not fill the ambient space. We also make an observation about the question of whether there exists a smooth, irreducible, non-degenerate, projective variety with secant deficiency delta > 8.

Partially Symmetric Tensors and the Non-defectivity of Secant Varieties of Products with a Projective Line as a Factor

Partially Symmetric Tensors and the Non-defectivity of Secant Varieties of Products with a Projective Line as a Factor

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

AbstractLet G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$ . We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$ . It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$ . Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$ . As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G-orbits in place of ${\mathcal O}_{\textsf{min}}$ .

Restricted secant varieties of Grassmannians

Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to $k$-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the Baur-Draisma-deGraaf Conjecture on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.

An algebraic geometry perspective for the estimation of the directions of arrival

Directions of Arrival for Uniform Linear Arrays represent a widely studied topic in many fields of signal processing, with large attention on computational complexity, estimation accuracy and noise rejection. In this article we study the mathematical model behind Directions of Arrival from the point of view of algebraic geometry, focusing on its relation with the rational normal curve. On this basis, we give a novel interpretation for the root-MUSIC algorithm, that is a widely adopted estimation approach in signal processing. Furthermore, we propose some novel estimation techniques. The first one is based on the computation of the points on the rational normal curve that minimize the distance from the linear subspace defined by the measured Directions of Arrival. The others require the study of the secant varieties of the rational normal curve and the minimization of the distance between the point of the Grassmannian defined by the signal subspace and a certain secant variety. The results obtained from simulations in a noisy scenario show that our estimators are statistically consistent. One algorithm outperforms root-MUSIC over a wide range of scenarios, especially in presence of few snapshots and low Signal to Noise Ratio.

Asymptotic syzygies of secant varieties of curves

We prove that the minimal free resolution of the secant variety of a curve is asymptotically pure. As a corollary, we show that the Betti numbers of converge to a normal distribution.

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.

Strength and slice rank of forms are generically equal

We prove that strength and slice rank of homogeneous polynomials of degree d ≥ 5 over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees 2 ≤ d ≤ 7 and d = 9.

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in Pn with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree n. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

Complete Singular Collineations and Quadrics

Abstract We construct wonderful compactifications of the spaces of linear maps and symmetric linear maps of a given rank as blowups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps.

Secant non-defectivity via collisions of fat points

The computation of dimensions of secant varieties of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. Non-defectivity can be proved via degenerations. In this paper, we use a technique that allows some of the base points to collapse together in order to deduce a new general criterion for non-defectivity. We apply this criterion to prove a conjecture by Abo and Brambilla: for c≥3 and d≥3, the Segre-Veronese embedding of Pm×Pn in bidegree (c,d) is non-defective.

On Severi varieties as intersections of a minimum number of quadrics

Let \({\mathscr{V}}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \({\mathscr{V}}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \({E_6}\)\((\mathbb R)\)) is the set-theoretic intersection of 15 quadrics.

A matryoshka structure of higher secant varieties and the generalized Bronowski's conjecture

In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized Kp,1 theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we prove a weak form of the generalized Bronowski's conjecture raised by C. Ciliberto and F. Russo that relates the identifiability for higher secant varieties to the geometry of tangential projections.

Distinguishing secant from cactus varieties

Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant varieties use only isolated reduced points. In particular, any secant variety is always contained in the respective cactus variety, and, except in a few initial cases, the inclusion is strict. It is known that lots of natural criteria that test membership in secant varieties are actually only tests for membership in cactus varieties. In this article, we propose the first techniques to distinguish actual secant variety from the cactus variety in the case of the Veronese variety. We focus on two initial cases, \(\kappa _{14}(\nu _d({\mathbb {P}}^n))\) and \(\kappa _{8,3}(\nu _d({\mathbb {P}}^n))\), the simplest that exhibit the difference between cactus and secant varieties. We show that for \(d\ge 5\), the component of the cactus variety \(\kappa _{14}(\nu _d({\mathbb {P}}^6))\) other than the secant variety \(\sigma _{14}(\nu _d({\mathbb {P}}^6))\) consists of degree d polynomials divisible by a \((d-3)\)rd power of a linear form. We generalize this description to an arbitrary number of variables. We present an algorithm for deciding whether a point in the cactus variety \(\kappa _{14}(\nu _d({\mathbb {P}}^n))\) belongs to the secant variety \(\sigma _{14}(\nu _d({\mathbb {P}}^n))\) for \(d\ge 6,\) \(n \ge 6\). We obtain similar results for the Grassmann cactus variety \(\kappa _{8,3}(\nu _d({\mathbb {P}}^n))\). Our intermediate results give also a partial answer to analogous problems for other cactus varieties and Grassmann cactus varieties to any Veronese variety.