Projective Space P3 Research Articles

Offline generator for digitally reconstructed radiographs of a commercial stereoscopic radiotherapy image-guidance system.

PurposeImage‐guided radiotherapy (IGRT) research sometimes involves simulated changes to patient positioning using retrospectively collected clinical data. For example, researchers may simulate patient misalignments to develop error detection algorithms or positioning optimization algorithms. The Brainlab ExacTrac system can be used to retrospectively “replay” simulated alignment scenarios but does not allow export of digitally reconstructed radiographs (DRRs) with simulated positioning variations for further analysis. Here we describe methods to overcome this limitation and replicate ExacTrac system DRRs by using projective geometry parameters contained in the ExacTrac configuration files saved for every imaged subject.MethodsTwo ExacTrac DRR generators were implemented, one with custom MATLAB software based on first principles, and the other using libraries from the Insight Segmentation and Registration Toolkit (ITK). A description of perspective projections for DRR rendering applications is included, with emphasis on linear operators in real projective space P3. We provide a general methodology for the extraction of relevant geometric values needed to replicate ExacTrac DRRs. Our generators were tested on phantom and patient images, both acquired in a known treatment position. We demonstrate the validity of our methods by comparing our generated DRRs to reference DRRs produced by the ExacTrac system during a treatment workflow using a manual landmark analysis as well as rigid registration with the elastix software package.ResultsManual landmarks selected between the corresponding DRR generators across patient and phantom images have an average displacement of 1.15 mm. For elastix image registrations, we found that absolute value vertical and horizontal translations were 0.18 and 0.35 mm on average, respectively. Rigid rotations were within 0.002 degrees.ConclusionCustom and ITK‐based algorithms successfully reproduce ExacTrac DRRs and have the distinctive advantage of incorporating any desired 6D couch position. An open‐source repository is provided separately for users to implement in IGRT patient positioning research.

Cubic graphs that cannot be covered with four perfect matchings

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is 3-edge-colourable, the rest of cubic graphs fall into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect matchings to cover their edges are particularly interesting as potential counterexamples to several profound and long-standing conjectures including the celebrated cycle double cover conjecture. However, so far they have been extremely difficult to find.In this paper we build a theory that describes coverings with four perfect matchings as flows whose flow values represent points and outflow patterns represent lines of a configuration of ten points and six lines spanned by four points of the 3-dimensional projective space P3(F2) in general position. This theory provides powerful tools for investigation of graphs that do not admit such a cover and offers a great variety of methods for their construction. As an illustrative example we produce a rich family of snarks (nontrivial cubic graphs with no 3-edge-colouring) that cannot be covered with four perfect matchings. The family contains all previously known graphs with this property.

Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space ℙ3 to a projective plane ℙ2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from ℙk to ℙh, with k > h ≥ 2. In those settings, the projective reconstruction of a scene consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.

Construction of symplectic vector bundles on projective space [formula omitted

The moduli spaces of symplectic vector bundles of arbitrary rank on projective space P3 are far from being well-understood. By now the only type of such bundles having satisfactory description are the so-called tame symplectic instantons. It is shown by U. Bruzzo, D. Markushevich and the first author in two papers from 2012 and 2016 that the moduli spaces of tame symplectic instantons are irreducible generically reduced algebraic spaces of dimension prescribed by the deformation theory. In the present paper we construct an infinite series of smooth irreducible moduli components of symplectic vector bundles of an arbitrary even rank 2r,r≥1, obtained by an iterative use of the monad construction applied to tame symplectic instantons. As a particular case we obtain an infinite series of irreducible moduli components of stable rank 2 vector bundles on P3. We show that this series contains as a subseries a large part of an infinite series of moduli components constructed by the authors and S. Tikhomirov in 2019. We also prove that, for any integers n,r, where r≥1 and n≥r+147, there exists a moduli component, not necessarily unique, of our series such that symplectic bundles from this component have rank 2r and second Chern class n.

Construction of Stable Rank 2 Bundles on ℙ3 Via Symplectic Bundles

In this article we study the Gieseker–Maruyama moduli spaces ℬ(e, n) of stable rank 2 algebraic vector bundles with Chern classes c1 = e ∈ {−1, 0} and c2 = n ≥ 1 on the projective space ℙ3. We construct the two new infinite series Σ0 and Σ1 of irreducible components of the spaces ℬ(e, n) for e = 0 and e = −1, respectively. General bundles of these components are obtained as cohomology sheaves of monads whose middle term is a rank 4 symplectic instanton bundle in case e = 0, respectively, twisted symplectic bundle in case e = −1. We show that the series Σ0 contains components for all big enough values of n (more precisely, at least for n ≥ 146). Σ0 yields the next example, after the series of instanton components, of an infinite series of components of ℬ(0, n) satisfying this property.

On the Number of Vedernikov–Ein Irreducible Components of the Moduli Space of Stable Rank 2 Bundles on the Projective Space

We propose a method for finding the exact number of Vedernikov–Ein irreducible components of the first and second types in the moduli space M(0, n) of stable rank 2 bundles on the projective space P3 with Chern classes c1 = 0 and c2 = n ≥ 1. We give formulas for the number of Vedernikov–Ein components and find a criterion for their existence for arbitrary n ≥ 1.

On the Number of Incidences Between Points and Planes in Three Dimensions

We prove an incidence theorem for points and planes in the projective space P3 over any Field F, whose characteristic p ≠ 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.

A loop group method for Demoulin surfaces in the 3-dimensional real projective space

A natural Gauss map for a surface in the 3-dimensional real projective space P3 will be defined and called the first-order Gauss map. It will be shown that the first-order Gauss map is conformal if and only if it is a Demoulin surface, which is a special case among projective minimal surfaces. Moreover, it will be shown that the first-order Gauss map is Lorentz harmonic if and only if it is a Demoulin surface or a projective minimal coincidence surface. We also characterize the surfaces via a family of flat connections on the trivial bundle D×SL4R over a simply connected domain D in the Euclidean 2-plane.

Families of stable bundles of rank 2 with c 1 =−1 on the space ℙ3

We construct some infinite series of families of stable rank 2 vector bundles on the projective space ℙ3 with odd first Chern class c1 = −1 and arbitrary second Chern class c2 = 2n with n ≥ 2. They are distinct from the series of families of bundles which were constructed by Hartshorne in 1978. We conjecture that for n ≥ 3 these families lie in the irreducible components of the moduli space of stable bundles distinct from the components that include Hartshorne’s families. In this article we prove the conjecture for n = 3. In this case the scheme of moduli of stable rank 2 vector bundles with c1 = −1 and c2 = 6 on ℙ3 has at least two irreducible components.

Reducibility of the Moduli Space of Stable Rank 2 Reflexive Sheaves with Chern Classes c1 = −1, c2 = 4, c3 = 2 on Projective Space P³

We prove the reducibility of the moduli space M ref P³ (2; −1, 4, 2) of stable rank 2 re- flexive sheaves with Chern classes c₁ = −1, c₂ = 4, c₃ = 2 on projective space P³. This gives the first example of a reducible space in the series of moduli spaces of stable rank 2 reflexive sheaves with Chern classes c₁= −1, c₂ = 4, c₃ = 2m, m = 1, 2, 3, 4, 5, 6, 8. We find two components of the expected dimension 27 of this space and give their geometric description via the Serre construction.

On set-theoretic complete intersection monomial curves in ℙ3

In this paper, we prove that there are infinitely many set-theoretic complete intersection monomial curves $\bar C$ (l p , l q , ap + bq) in the projective space ℙ3 for every suitably chosen integers l, a, b. We also give explicitly the polynomials defining the corresponding surfaces.

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I * of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).

On the parameter dependence of the real cubic surfaces in arnold’s form

We consider the topological classification of cubic surfaces which are obtained as intersection of the sphere\(\mathbb{S}^3 \) with the algebraic variety defined by the zeroes of a homogeneous cubic polynomial in Arnold’s normal form. This classification is based on the parameters appearing in this normal form, obtaining a correspondence between the parameters of the surface and its topological type. General classifications of cubic surfaces are made in the projective space ℙ3(ℝ), but our method, based on a very simple combinatorial procedure is easier to implement in\(\mathbb{S}^3 \). We split the cubic surfaces parameter space into ten equivalence classes.

On the measurability of the conic sections’ family in the projective spaceP 3 3

In this paper we prove that the non degenerate conic sections’ family in the projective spaceP 3 is measurable.

An irreducible Heegaard diagram of the real projective 3‐space p3

We give a genus 3 Heegaard diagram H of the real projective space p3, which has no waves and pairs of complementary handles. So Negami′s result that every genus 2 Heegaard diagram of p3 is reducible cannot be extended to Heegaard diagrams of p3 with genus 3.

Determinantal Representation of a Class of Surfaces

We present a description of determinantal representation of certain quartic surfaces of the projective space P 3(K), 3-dimensional over a field K, char (K) ≠ 2.

Über verallgemeinerte W-Kongruenzen im projektiven Raum P3, deren eine Fokalfläche entartet

In the real projective space ℙ3 a W-congruence has the characteristic property that its second osculating space has a dimension d(2)≤4 [4]. In this paper we extend the concept of W-congruences to the case of not hyperbolic congruences by using the preceding property and we find all W-congruences with a degenerated focal surface.

Zur Kr�mmungsverwandtschaft Von Zwangl�ufen in Affinen CK ? Ebenen I

In part I we have studied a map of ”osculating elements” of an affine Cayley-Klein (CK-) plane into the Lie algebra A4(ɛ2) of the aequiform transformations A4(ɛ2) of the given plane A2(ℜ, ɛ2). If we use the real projective space P3(ℜ) over A4(ɛ2) each osculating element defines a straight line in P3(ℜ). We now give a one parameter motion in A4(ɛ2) and study second order properties and their analogon in the Lie algebra and P3(ℜ), respectively. We show that the wellknown relationship between the points of the moving frame and the osculating circles of the point paths in the fixed frame may be interpreted as part of a quadratic map Γ of certain straight Lines of P3(ℜ). An analogous result holds for the curvature of pairs of envelopes; the mapV induced in P3(ℜ) than is contained in a cubic relationship of straight lines.

SMOOTHNESS OF THE MODULI SCHEME OF INSTANTON VECTOR BUNDLES WITHc1=0,c2=nONP3

The scheme M(P3, n) of moduli of mathematical instanton bundles with Chern classes c1 = 0, c2 = n ≥ 1, is studied on projective space P3 over the field C. It is proved that M(P3, n) is a smooth equidimensional scheme of dimension 8n – 3. Bibliography: 7 titles.

Singular surfaces and the rigidity of ?3

In this paper we give two elementary proofs for the rigidity of the complex projective space ℙ3 with respect to global deformations.