Segre Varieties Research Articles

Nash conditional independence curve

We study the Spohn conditional independence (CI) variety CX of an n-player game X for undirected graphical models on n binary random variables consisting of one edge. For a generic game, we show that CX is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety (P1)n−2×P3 and we give an explicit formula for its degree and genus. We prove two universality theorems for CX: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in Rm defined by at most m−1 polynomials is isomorphic to an affine open subset of CX for some game X.

Terracini locus for three points on a Segre variety

Terracini locus for three points on a Segre variety

Terracini Loci of Segre Varieties

Fix a format (n1+1)×⋯×(nk+1), k>1, for real or complex tensors and the associated multiprojective space Y. Let V be the vector space of all tensors of the prescribed format. Let S(Y,x) denote the set of all subsets of Y with cardinality x. Elements of S(Y,x) are associated to rank 1 decompositions of tensors T∈V. We study the dimension δ(2S,Y) of the kernel at S of the differential of the associated algebraic map S(Y,x)→PV. The set T1(Y,x) of all S∈S(Y,x) such that δ(2S,Y)>0 is the largest and less interesting x-Terracini locus for tensors T∈V. Moreover, we consider the one (minimally Terracini) such that δ(2A,Y)=0 for all A⊈S. We define and study two different types of subsets of T1(Y,x) (primitive Terracini and solution sets). A previous work (Ballico, Bernardi, and Santarsiero) provided a complete classification for the cases x=2,3. We consider the case x=4 and several extremal cases for arbitrary x.

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.

Curvilinear subschemes of Segre varieties and the cactus rank

We study zero-dimensional linearly dependent subschemes W of the Segre variety with deg(W)≤5. If W is connected and curvilinear with arbitrary degree we give a strong restriction on the number of factors of the concise Segre containing W.

Foliations by Curves on Three-Dimensional Segre Varieties

Foliations by Curves on Three-Dimensional Segre Varieties

Segre-degenerate points form a semianalytic set

We prove that the set of Segre-degenerate points of a real-analytic subvariety X X in C n {\mathbb {C}}^n is a closed semianalytic set. It is a subvariety if X X is coherent. More precisely, the set of points where the germ of the Segre variety is of dimension k k or greater is a closed semianalytic set in general, and for a coherent X X , it is a real-analytic subvariety of X X . For a hypersurface X X in C n {\mathbb {C}}^n , the set of Segre-degenerate points, X [ n ] X_{[n]} , is a semianalytic set of dimension at most 2 n − 4 2n-4 . If X X is coherent, then X [ n ] X_{[n]} is a complex subvariety of (complex) dimension n − 2 n-2 . Example hypersurfaces are given showing that X [ n ] X_{[n]} need not be a subvariety and that it also need not be complex; X [ n ] X_{[n]} can, for instance, be a real line.

Real-analytic coordinates for smooth strictly pseudoconvex CR-structures

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieties in terms of their $1$-jets. We also express this condition in equivalent terms for a Fefferman type determinant

Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

Apolarity, border rank, and multigraded Hilbert scheme

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyze its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.

Open -ranks with respect to Segre and Veronese varieties

Let be an integral and non-degenerate variety. Recall (A. Bialynicki-Birula, A. Schinzel, J. Jelisiejew and others) that for any the open rank is the minimal positive integer such that for each closed set there is a set with and , where denotes the linear span. For an arbitrary we give an upper bound for in terms of the upper bound for when is a point in the maximal proper secant variety of and a similar result using only points with submaximal border rank. We study when is a Segre variety (points with -rank and ) and when is a Veronese variety (points with -rank or with border rank ).

Asymptotics of degrees and ED degrees of Segre products

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties.We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation.Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface.

Complete intersections of quadrics and complete intersections on Segre varieties with common specializations

We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product \mathbf{P}^1\times \mathbf{P}^k\hookrightarrow \mathbf{P}^{2k+1} can belong to the same Hilbert scheme. For k=2 there is a classical example; it comes from K3 surfaces in projective 5 -space that degenerate into a hypersurface on the Segre threefold. We show that for k\geq 3 there is only one more example. It turns out that its (connected) Hilbert scheme has at least two irreducible components. We investigate the corresponding local moduli problem.

Extremal discs and Segre varieties for real-analytic hypersurfaces in $\mathbb C^2$

We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.

Vanishing Hachtroudi Curvature and Local Equivalence to the Heisenberg Pseudosphere

To any completely integrable second-order system of real or complex partial differential equations: $$\begin{aligned} y_{x^{k_1}x^{k_2}} = F_{k_1,k_2} \left( x^1,\dots ,x^n,\,y,\,y_{x^1},\ldots ,y_{x^n} \right) \end{aligned}$$ with $$1 \leqslant k_1,\, k_2 \leqslant n$$ and with $$F_{ k_1, k_2} = F_{ k_2, k_1}$$ in $$\underline{n \geqslant 2}$$ independent variables $$(x^1, \ldots , x^n)$$ and in one dependent variable y, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface M in $$\mathbb {C}^n$$ ( $$n \geqslant 3$$ ) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct $$\mathbb {C}^2$$ -case, this then characterizes in an effective way those hypersurfaces of $$\mathbb {C}^{n+1}$$ in higher complex dimension $$n + 1\geqslant 3$$ that are locally biholomorphic to a piece of the $$(2n + 1)$$ -dimensional Heisenberg quadric, without any special assumption on their defining equations.

Linearly dependent subsets of Segre varieties

We study the linear algebra of finite subsets S of a Segre variety X. In particular we classify the pairs (S, X) with S linear dependent and $$\#(S)\le 5$$ . We consider an additional condition for linear dependent sets, i.e. that no two of their points are contained in a line of X, and get far better lower bounds for $$\#(S)$$ in term of the dimension and number of the factors of X. In this discussion and in the classification of the case $$\#(S)= 5$$ , $$X\cong {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$ we use the rational normal curves contained in X.

The symmetric representation of lines in [formula omitted

Let F be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V=F3⊗F3 of 3 × 3 matrices over F, and let G≤PGL(V) be the setwise stabiliser of the corresponding Segre variety S3,3(F) in the projective space PG(V). The G-orbits of lines in PG(V) were determined by the first author and Sheekey as part of their classification of tensors in F2⊗V in [15]. Here we solve the related problem of classifying those line orbits that may be represented by symmetric matrices, or equivalently, of classifying the line orbits in the F-span of the Veronese variety V3(F)⊂S3,3(F) under the natural action of K=PGL(3,F). Interestingly, several of the G-orbits that have symmetric representatives split under the action of K, and in many cases this splitting depends on the characteristic of F. Although our main focus is on the case where F is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where F is an algebraically closed field, or the field of real numbers. The corresponding orbit sizes and stabiliser subgroups of K are also determined in the case where F is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in PG(2,F), or equivalently, of pairs of ternary quadratic forms over F.

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.

Secant varieties of the varieties of reducible hypersurfaces in [formula omitted

Given the space V=P(d+n−1n−1)−1 of forms of degree d in n variables, and given an integer ℓ>1 and a partition λ of d=d1+⋯+dr, it is in general an open problem to obtain the dimensions of the (ℓ−1)-secant varieties σℓ(Xn−1,λ) for the subvariety Xn−1,λ⊂V of hypersurfaces whose defining forms have a factorization into forms of degrees d1,…,dr. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of σℓ(Xn−1,λ) for any choice of parameters n,ℓ and λ. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., r=2), we also relate this problem to a conjecture by Fröberg on the Hilbert function of an ideal generated by general forms.

Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

Making use of the `Veldkamp blow-up' recipe, introduced by Saniga and others (Ann. Inst. H. Poincar\' e D2 (2015) 309) for binary Segre varieties, we study geometric hyperplanes and Veldkamp lines of Segre varieties $S_k(3)$, where $S_k(3)$ stands for the $k$-fold direct product of projective lines of size four and $k$ runs from 2 to 4. Unlike the binary case, the Veldkamp spaces here feature also non-projective elements. Although for $k=2$ such elements are found only among Veldkamp lines, for $k \geq 3$ they are also present among Veldkamp points of the associated Segre variety. Even if we consider only projective geometric hyperplanes, we find four different types of non-projective Veldkamp lines of $S_3(3)$, having 2268 members in total, and five more types if non-projective ovoids are also taken into account. Sole geometric and combinatorial arguments lead to as many as 62 types of projective Veldkamp lines of $S_3(3)$, whose blowing-ups yield 43 distinct types of projective geometric hyperplanes of $S_4(3)$. As the latter number falls short of 48, the number of different large orbits of $2 \times 2 \times 2 \times 2$ arrays over the three-element field found by Bremner and Stavrou (Lin. Multilin. Algebra 61 (2013) 986), there are five (explicitly indicated) hyperplane types such that each is the merger of two different large orbits. Furthermore, we single out those 22 types of geometric hyperplanes of $S_4(3)$, featuring 7 176 640 members in total, that are in a one-to-one correspondence with the points lying on the unique hyperbolic quadric $\mathcal{Q}_0^{+}(15,3) \subset {\rm PG}(15,3) \subset \mathcal{V}(S_4(3))$; and, out of them, seven ones that correspond bijectively to the set of 91 840 generators of the symplectic polar space $\mathcal{W}(7,3) \subset \mathcal{V}(S_3(3))$. For $k=3$ we also discuss embedding of the binary Veldkamp space into the ternary one.