Higher Dimensional Projective Spaces Research Articles

Problems and related results in algebraic vision and multiview geometry

AbstractThis article is a survey of results in algebraic vision and multiview geometry. The starting point is the study of projective algebraic varieties which are critical for scene reconstruction. Initially studied for reconstructing static scenes in three-dimensional spaces, these critical loci are later investigated for dynamic and segmented scenes in higher-dimensional projective spaces. The formal analysis of the ideals of critical loci employs Grassmann tensors, introduced as crucial tools for determining these ideals and aiding the reconstruction process away from critical loci. A long-term goal of the authors with other co-authors involves two main aspects: firstly studying properties of Grassmann tensors, as rank, multi-rank and core, along with the varieties that parameterize these tensors; concurrently conducting an analysis of families of critical loci in various scenarios.

REAL TOPOLOGICAL HOCHSCHILD HOMOLOGY OF SCHEMES

AbstractWe prove that real topological Hochschild homology $\mathrm {THR}$ for schemes with involution satisfies base change and descent for the ${\mathbb {Z}/2}$ -isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher-dimensional projective spaces.

An algorithm for counting arcs in higher-dimensional projective space

An n-arc in (k−1)-dimensional projective space is a set of n points so that no k lie on a hyperplane. In 1988, Glynn gave a formula to count n-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count n-arcs in the projective plane for n≤10. In this paper, we determine a formula to count n-arcs in projective 3-space. We then use this formula to give exact expressions for the number of n-arcs in P3(Fq) for n≤7, which are polynomial in q for n≤6 and quasipolynomial in q for n=7. Lastly, we generalize to higher-dimensional projective space.

Resolving sets for higher dimensional projective spaces

Lower and upper bounds on the size of resolving sets for the point-hyperplane incidence graph of the finite projective space PG(n,q) are presented.

On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].

Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space

A simplex in a projective space of dimension n is expressed by a matrix of order n + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Mobius pair along with the reference simplex. A pair of mutually inscribed, circumscribed tetrahedrons is called a Mobius pair. The existence of such pairs of simplexes in higher-dimensional (odd) projective spaces is already established. In the present study an existence of an infinite chain of simplexes in a five-dimensional projective space is established where any two simplexes from the chain form a Mobius pair in some order of their vertices. This is studied with the help of powers of a block Toeplitz matrix. Also, attempt has been made to generalize this result to 2n + 1-dimensional projective space.

Singular solutions in soft limits

A generalization of the scattering equations on X (2, n), the configuration space of n points on ℂℙ1, to higher dimensional projective spaces was recently introduced by Early, Guevara, Mizera, and one of the authors. One of the new features in X (k, n) with k > 2 is the presence of both regular and singular solutions in a soft limit. In this work we study soft limits in X (3, 7), X (4, 7), X (3, 8) and X (5, 8), find all singular solutions, and show their geometrical configurations. More explicitly, for X (3, 7) and X (4, 7) we find 180 and 120 singular solutions which when added to the known number of regular solutions both give rise to 1 272 solutions as it is expected since X (3, 7) ∼ X (4, 7). Likewise, for X (3, 8) and X (5, 8) we find 59 640 and 58 800 singular solutions which when added to the regular solutions both give rise to 188 112 solutions. We also propose a classification of all configurations that can support singular solutions for general X (k, n) and comment on their contribution to soft expansions of generalized biadjoint amplitudes.

Orbits in [formula omitted] and equivariant quantum cohomology

We compute the GLr+1-equivariant Chow class of the GLr+1-orbit closure of any point (x1,…,xn)∈(Pr)n in terms of the rank polytope of the matroid represented by x1,…,xn∈Pr. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each d×n matrix M an n-ary operation [M]ħ on the small equivariant quantum cohomology ring of Pr, which is the n-ary quantum product when M is an invertible matrix. We prove that M↦[M]ħ is a valuative matroid polytope association.Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree 2r+1 hypersurface in Pr, generalizing the known case of quintic plane curves.

Scattering equations: from projective spaces to tropical grassmannians

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors {mathrm{s}}_{a_1{a}_2dots {a}_k} subject to a ‘massless’ condition {mathrm{s}}_{a_1{a}_2dots {a}_{k-2}bb}=0 and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.

On the distribution of Weierstrass points on Gorenstein quintic curves

Abstract This paper is concerned with developing a technique to compute in a very precise way the distribution of Weierstrass points on the members of any 1-parameter family Ca, a ∈ C , of Gorenstein quintic curves with respect to the dualizing sheaf K C a . The nicest feature of the procedure is that it gives a way to produce examples of existence of Weierstrass points with prescribed special gap sequences, by looking at plane curves or, more generally, to subcanonical curves embedded in some higher dimensional projective space.

Duality of planar and spacial curves: new insight

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for a regular behavior of the equiclassical stratification. We also discuss duality of curves in higher-dimensional projective spaces and in Grassmannians with focus on similar questions of the regularity of equiclassical families of spacial curves.

Postulation of General Unions of Lines and Double Points in a Higher Dimensional Projective Space

We prove that a general union $X\subset \mathbb {P}^{r}$ , r ≥ 6, of lines and double points has the expected postulation with respect to the hypersurfaces of degree ≥ 4.

Symmetric topological complexity as the first obstruction in Goodwillie’s Euclidean embedding tower for real projective spaces

This paper explains why Goodwillie-Weiss calculus of embeddings can offer new information about the Euclidean embedding dimension of P m only for m � 15. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential—but critical—high-order obstructions in the corresponding Taylor towers. For m � 16, the relation TC S (P m ) � n is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding P m � R n . A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis’ BP-approach to the immersion problem of P m . A form of the Euler class viewpoint is applied to show TC S (P 3 ) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber’s work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber’s lead, this concept is connected to TC S (C(S)), the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P 3 .

Interactive Dimensional Synthesis and Motion Design of Planar 6R Single-Loop Closed Chains via Constraint Manifold Modification

In this paper, we present an interactive, visual design approach for the dimensional synthesis of planar 6R single-loop closed chains for a given rational motion using constraint manifold modification. This approach is implemented in an interactive software tool that provides mechanism designers with an intuitive way to determine the parameters of planar mechanisms, and in the process equips them with an understanding of the design process. The theoretical foundation of this work is based on representing planar displacements with planar quaternions, which can be seen as points in a special higher dimensional projective space (called the image space), and on formulating the kinematic constraints of closed chains as algebraic surfaces in the image space. Kinematic constraints under consideration limit the positions and orientation of the coupler in its workspace. In this way, a given motion of a mechanism in the Cartesian space maps to a curve in the image space that is limited to stay within the bounds of the algebraic surfaces. Thus, the problem of dimensional synthesis is reduced to determine the parameters of equations that describe algebraic surfaces. We show that the interactive approach presented here is general in nature, and can be easily used for the dimensional synthesis of any mechanism for which kinematic constraints can be expressed algebraically. The process of designing is fast, intuitive, and especially useful when a numerical optimization based approach would be computationally demanding and mathematically difficult to formulate. This simple approach also provides a basis for students and early designers to learn and understand the process of mechanism design by simple geometric manipulations.

A new construction of caps

In this paper we present a general method to construct caps in higher-dimensional projective spaces. As an application, for q ≥ 8 even we obtain caps in PG(5,q) larger than the caps known so far, and a new class of caps of size ( q + 1 ) ( q 2 + 3 ) for q ≥ 7 odd.

GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY

Abstract

Linear series over real and $p$-adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve C of genus g to P 1 with prescribed ramification also yields weaker results when working over the real numbers or p-adic fields. Specifically, let k be such a field: we see that given g, d, n, and e 1 ,..e n satisfying Σ ι (e i -1) = 2d - 2 - g, there exists smooth curves C of genus g together with points P 1 ,..., P n such that all maps from C to P 1 can, up to automorphism of the image, be defined over k. We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case C = P 1 , n = 3, and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

Representation theory and ADHM-construction on quaternion symmetric spaces

We determine all irreducible homogeneous bundles with anti-selfdual canonical connections on compact quaternion symmetric spaces. To deform the canonical connections, we give a relation between the representation theory and the theory of monads on the twistor space. The moduli spaces are described via the Bott-Borel-Weil Thereom. The Horrocks bundle is also generalized to higher-dimensional projective spaces.

A remark on the geometry of elliptic scrolls and bielliptic surfaces

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC r in ℙ r such thatC r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.