Real Quadrics Research Articles

Rationality of even-dimensional intersections of two real quadrics

We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.

Moment Maps and Galois Orbits in Quantum Information Theory

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

On a classical correspondence of real surfaces

This paper is devoted to the classical correspondence between real surfaces of degree 8 which are complete intersections of three real quadrics, and real surfaces which are two-sheeted coverings of the corresponding pencils of quadrics with branching along the curves of degenerate quadrics.

On intersections of two real quadrics

This paper is devoted to describing properties of complete intersections of two real projective quadrics. For brevity we call such varieties biquadrics. One of the main sections is devoted to real projective spaces on real biquadrics. In another main section we study the topology of the real parts of biquadrics. The other sections are auxiliary.

Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ O ( p , q ) on Real Quadrics

Let $$X=G/P$$ be a real projective quadric, where $$G=O(p,\,q)$$ and P is a parabolic subgroup of G. Let $$(\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}$$ be the family of (smooth) representations of G induced from the characters of P. For $$(\lambda ,\, \epsilon ),\, (\mu ,\, \eta )\in {\mathbb {C}}\times \{\pm \},$$ a differential operator $$\mathbf D_{(\mu ,\eta )}^\mathrm{reg}$$ on $$X\times X,$$ acting G-covariantly from $${\mathcal {H}}_{\lambda ,\epsilon } \otimes {\mathcal {H}}_{\mu , \eta }$$ into $${\mathcal {H}}_{\lambda +1,-\epsilon } \otimes {\mathcal {H}}_{\mu +1, -\eta }$$ is constructed.

Mutants of compactified representations revisited

We show that the mutants of compactified representations constructed by Franz and Puppe can be written as intersections of real quadrics involving division algebras and as generalizations of polygon spaces. We also show that these manifolds are connected sums of products of spheres.

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Let W be a linear system of quadrics on the real projective space \mathbb R P^n and X be the base locus of that system (i.e. the common zero set of the quadrics in W ). We prove a formula relating the topology of X to the one of the discriminant locus \Sigma_W (i.e. the set of singular quadrics in W ). The set \Sigma_W equals the intersection of W with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of W ) and we set \{\Sigma_W^{(r)}\}_{r\geq 1} for its singular point stratification, i.e. \Sigma_W^{(1)}=\Sigma_W and \Sigma_W^{(r)}=\textrm{Sing}\big( \Sigma_W^{(r-1)}\big) . With this notation, for a generic W the mentioned formula writes: b(X) \leq b(\mathbb R P^n)+ \sum_{r \geq 1}b(\mathbb{P}\Sigma_W^{(r)}). In the general case a similar formula holds, but we have to replace each b(\mathbb{P}\Sigma_W^{(r)}) with \frac{1}{2}b(\Sigma_\epsilon^{(r)}) , where \Sigma_\epsilon equals the intersection of the discriminant hypersurface with the unit sphere on the translation of W in the direction of a small negative definite form. Each \Sigma_\epsilon^{(r)} is a determinantal variety on the sphere S^{k-1} defined by equations of degree at most n+1 (here k denotes the dimension of W ); we refine Milnor's bound, proving that for such affine varieties b(\Sigma_\epsilon^{(r)})\leq O(n)^{k-1} . Since the sum in the above formulas contains at most O(k)^{1/2} terms, as a corollary we prove that if X is any intersection of k quadrics in \mathbb R P^n then the following sharp estimate holds: b(X) \leq O(n)^{k-1}. This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape O(n)^{2k+2} ).

Big Polygon Spaces

We study a new class of compact orientable manifolds, called big polygon spaces. They are intersections of real quadrics and related to polygon spaces, which appear as their fixed point set under a canonical torus action. What makes big polygon spaces interesting is that they exhibit remarkable new features in equivariant cohomology: The Chang-Skjelbred sequence can be exact for them and the equivariant Poincare pairing perfect although their equivariant cohomology is never free as a module over the cohomology ring of BT. More generally, big polygon spaces show that a certain bound on the syzygy order of the equivariant cohomology of compact orientable T-manifolds obtained by Allday, Puppe and the author is sharp.

BIHARMONIC CURVES INTO QUADRICS

AbstractWe develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves in surfaces defined by a polynomial equation: In particular, we use it to give a complete classification of biharmonic curves in real quadrics of the three-dimensional Euclidean space.

The total Betti number of the intersection of three real quadrics

Abstract The classical Barvinok bound for the sum of the Betti numbers of the intersection X of three quadrics in ℝPn says that there exists a natural number a such that b(X) ≤ n3a. We improve this bound proving the inequality b(X) ≤ n(n+1). Moreover we show that this bound is asymptotically sharp as n goes to infinity.

Real space quadrics and μ-bases

Abstract In Euclidean 3-space, a quadric surface is the zero set of a quadratic equation in three variables. Its projective closure can be given as the closure of the image of a rational parametrization P : R 2 → R 4 where P maps the parameters ( s , t ) ∈ R 2 to the tuple ( a , b , c , d ) ∈ R 4 and a, b, c, d are linearly independent quadratic polynomials, with gcd(a, b, c, d) = 1. This paper provides an algorithm to classify the type of quadric surface, and identify the normal forms solely based on the parametrization of the quadric surface.

Real four-dimensional GM-triquadrics

Nonsingular intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensional triquadrics. Necessary and sufficient conditions for a real four-dimensional triquadric to be a GM-variety are established.

On the number of invariant conics for the polynomial vector fields defined on quadrics

The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X=(P,Q,R) in R3 whose flow leaves Q invariant. If m1 = degree P, m2 = degree Q and m3 = degree R, we say that m=(m1,m2,m3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that results from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non-degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non-degenerate conics of the vector field X depending on each quadric Q and of the degrees m1, m2 and m3 of X.

Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group with fibre a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.

Real -triquadrics

We consider non-singular intersections of three real quadrics. For brevity, such varieties are called real triquadrics. We prove criteria for the maximality of triquadrics. These criteria are based on a connection between triquadrics and plane curves.

Hamiltonian-minimal Lagrangian submanifolds in toric varieties

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebrated Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism. In the previous works the authors defined and studied a family of H-minimal Lagrangian submanifolds in complex space arising from intersections of Hermitian quadrics. Here we extend this construction to define H-minimal submanifolds in toric varieties.

Cohomology of real four-dimensional triquadrics

We consider non-singular intersections of three real six-dimensional quadrics. They are referred to for brevity as real four-dimensional triquadrics. We calculate the dimensions of their cohomology spaces with coefficients in the field of two elements.

Convex Pencils of Real Quadratic Forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ℝPn (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere Sn; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)≤2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number bk(X)≤2(k+2).

Cohomology of real three-dimensional triquadrics

We consider non-singular intersections of three real five-dimensional quadrics. They are referred to for brevity as real three-dimensional triquadrics. We calculate the dimensions of the cohomology spaces of triquadrics with coefficients in the field of two elements.

On the number of components of a three-dimensional maximal intersection of three real quadrics

We consider non-singular intersections of three real five-dimensional quadrics. For brevity they are referred to as real three-dimensional triquadrics. We prove the existence of real three-dimensional -triquadrics with components, where is any integer in the range .