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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." [ABSTRACT FROM AUTHOR]

In this paper, we study certain variational aspects of the volume product functional restricted to the space of small projective deformations of a fixed convex body. In doing so, we provide a short proof of a theorem by Klartag: a strong version of the slicing conjecture implies the non-symmetric Mahler conjecture. We also exhibit an interesting family of critical convex bodies in dimension |$2$| containing saddle points for this functional. [ABSTRACT FROM AUTHOR]

Given X$X$, a K3 surface admitting a symplectic automorphism τ$\tau$ of order 4, we describe the isometry τ∗$\tau ^*$ on H2(X,Z)$H^2(X,\mathbb {Z})$. Having called Z∼$\tilde{Z}$ and Y∼$\tilde{Y}$, respectively, the minimal resolutions of the quotient surfaces Z=X/τ2$Z=X/\tau ^2$ and Y=X/τ$Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps X→Z∼,X→Y∼$X\rightarrow \tilde{Z},\ X\rightarrow \tilde{Y}$ and Y∼→Z∼$\tilde{Y}\rightarrow \tilde{Z}$: With this knowledge, we are able to give a lattice‐theoretic characterization of Z∼$\tilde{Z}$, and find the relation between the Néron–Severi lattices of X,Z∼$X,\tilde{Z}$ and Y∼$\tilde{Y}$ in the projective case. We also produce three different projective models for X,Z∼$X,\tilde{Z}$ and Y∼$\tilde{Y}$, each associated to a different polarization of degree 4 on X$X$. [ABSTRACT FROM AUTHOR]

A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean ×$\times$ Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. [ABSTRACT FROM AUTHOR]

We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension‐2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on a strong blocking set. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length n$n$ with the property that for any three distinct codewords there is a coordinate where they all have distinct values. Over the finite field F3$\mathbb {F}_3$, we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n$n$ has size at most 3n/4.55$3^{n/4.55}$, improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n$n$ and size at least 139/5n/4$\frac{1}{3}{\left(9/5 \right)}^{n/4}$, thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension‐2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to F3$\mathbb {F}_3$, we obtain linear trifferent codes of size at least 323n/312$3^{23n/312}$, improving the current best explicit construction that has size 3n/112$3^{n/112}$. [ABSTRACT FROM AUTHOR]

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples. [ABSTRACT FROM AUTHOR]

We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data. [ABSTRACT FROM AUTHOR]

We prove that for an embedded unstable one-sided minimal hypersurface of the |$(n+1)$| -dimensional real projective space, the Morse index is at least |$n+2$|⁠ , and this bound is attained by the cubic isoparametric minimal hypersurfaces. We also show that there exist closed embedded two-sided minimal surfaces in the 3-dimensional real projective space of each odd index by computing the index of the Lawson surfaces. [ABSTRACT FROM AUTHOR]

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds—that is, independent of the characteristic—on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves. [ABSTRACT FROM AUTHOR]

We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the variety? We exhibit some conditions under which this question has an affirmative answer. We primarily consider ample bundles on projective spaces and Hirzebruch surfaces. We also show that Seshadri constants of ample vector bundles can be arbitrarily small. [ABSTRACT FROM AUTHOR]

In their research, the authors actively exploit different branches of geometry. For geometric constructions, computer algebra approaches and systems are used. Currently, we are interested in computer geometry, more specifically, the implementation of computer graphics. The use of the projective space and homogeneous coordinates has actually become a standard in modern computer graphics. In other words, the problem is reduced to the application of analytic projective geometry. The authors failed to find a computer algebra system that could implement projective geometry in its entirety. Therefore, it was decided to partially implement computer algebra for visualization of algebraic relations. For this purpose, the Asymptote system was employed. [ABSTRACT FROM AUTHOR]

In this work, we describe the moduli of triples of points in quaternionic projective space which define uniquely the congruence classes of such triples relative to the action of the isometry group of quaternionic hyperbolic space H Q n. To solve this problem, we introduce some basic invariants of triples of points in quaternionic hyperbolic geometry. In particular, we define quaternionic analogues of the Goldman invariants for mixed configurations of points introduced by him in complex hyperbolic geometry. [ABSTRACT FROM AUTHOR]

We continue the study of Terracini loci formed by x points of a variety embedded in a projective space. Our main results are a refined study of Terracini loci arising from linear projections, the description of the maximal x with a non-empty Terracini locus for Hirzebruch surfaces, and the maximal "weight", "corank", or "defect" in several cases. For low x, we even show which defects can occur. [ABSTRACT FROM AUTHOR]

Let π:Z→Pn−1$\pi :Z\rightarrow \mathbb {P}^{n-1}$ be a general minimal n$n$‐fold conic bundle with a hypersurface BZ⊂Pn−1$B_Z\subset \mathbb {P}^{n-1}$ of degree d$d$ as discriminant. We prove that if d≥4n+1$d\ge 4n+1$, then −KZ$-K_Z$ is not pseudo‐effective, and that if d=4n$d = 4n$, then none of the integral multiples of −KZ$-K_{Z}$ is effective. Finally, we provide examples of smooth unirational n$n$‐fold conic bundles π:Z→Pn−1$\pi :Z\rightarrow \mathbb {P}^{n-1}$ with a discriminant of arbitrarily high degree. [ABSTRACT FROM AUTHOR]

We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H -Hopf–Galois extensions B ⊆ A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H -comodule algebra automorphism of A. We work out in detail the U (1) -extension O (C P q n − 1) ⊆ O (S q 2 n − 1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K -theory of the base algebra O (C P q n − 1). [ABSTRACT FROM AUTHOR]

Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P (1 , 1 , 4 , 6) . [ABSTRACT FROM AUTHOR]

We prove that quotients of the Fulton-MacPherson compactification of configuration spaces of smooth projective varieties of dimension >1 by permutation groups have canonical singularities. [ABSTRACT FROM AUTHOR]

The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range in which the graded Betti numbers of P n 1 × P n 2 embedded by O P n 1 × P n 2 (d 1 , d 2) are non-zero. [ABSTRACT FROM AUTHOR]

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 (P 3) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective. [ABSTRACT FROM AUTHOR]

In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector | T 〉 to the state vector | Ψ 〉 and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector | T 〉 to the state vector | Ψ 〉 , orthogonal to | T 〉 and | Ψ 〉 and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian. [ABSTRACT FROM AUTHOR]

We describe spaces that are dual to some subspaces of the space C∞(D) relative to the inductive limits, where inductive limits, where D ⊂ R p is a bounded convex domain. For any logarithmically convex space of positive numbers M = M k , k ∈ Z + p we introduce the normed space C D , M of functions f ∈ C ∞ D and prove that the Fourier-Laplace transform establishes a topological isomorphism between the strongly dual space and the projective limit of the normed spaces. [ABSTRACT FROM AUTHOR]

We study the blowing-up X ~ of a smooth projective variety X along a smooth center B that is equipped with a projective bundle structure over a variety Z. If the Picard number ρ (X) is 1 and dim ⁡ X is at most 4, we classify all such pairs (X , B). If X is a projective space P n (n ≥ 5) and dim ⁡ B is 2, we show that B is a linear subspace in X. [ABSTRACT FROM AUTHOR]

We consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to K3 surfaces andFano threefolds. In particular,we considerK3 surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pullback of plane curves of degree k ≥ 7 lies on a unique K3 surface. If k ≤ 6, then the general such curve is instead extendable to a higher dimensional variety. In the cases k =4, 5, 6, this gives the existence of singular index-k Fano varieties of dimensions 8, 5, 3, genera 17, 26, 37, and indices 6, 3, 1, respectively. For k =6, we recover the Fano varietyP( 3, 1, 1, 1), one of only two Fano threefolds with canonicalGorenstein singularities with the maximal genus 37 found by Prokhorov.We show that the latter variety is not further extendable. For k =4 and 5, these Fano varieties have been identified by Totaro. We also study the extensions of smooth degree-2 sections of K3 surfaces of genus 3. Inall these cases, we compute the corank of theGauss--Wahl maps of the curves under consideration. Finally, we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties. [ABSTRACT FROM AUTHOR]

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDE s) associated with the quantum cohomology of P 1 -bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P 1 -bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17] , called Borel (α , β) -multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P 1 -bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDE s. As an example, we show how to integrate the qDE of blow-up of P 2 at one point via Borel multitransforms of solutions of the qDE of P 1. [ABSTRACT FROM AUTHOR]

We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior. [ABSTRACT FROM AUTHOR]

The Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus‐invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials. [ABSTRACT FROM AUTHOR]

The Emergent String Conjecture of Lee, Lerche, and Weigand holds that every infinite-distance limit in the moduli space of a quantum gravity represents either a decompactification limit or an emergent string limit in some duality frame. Within the context of 5d supergravities coming from M-theory compactifications on Calabi-Yau threefolds, we find evidence for this conjecture by studying (a) the gauge couplings and (b) the BPS spectrum, which is encoded in the Gopakumar-Vafa invariants of the threefold. In the process, we disuss a testable geometric consequence of the Emergent String Conjecture, and we verify that it is satisfied in all complete intersection Calabi-Yau threefolds in products of projective spaces (CICYs). [ABSTRACT FROM AUTHOR]

Conjecture |${{\mathcal{O}}}$| and Gamma conjectures for quantum cohomology of Fano manifolds were proposed by Galkin, Golyshev, and Iritani [ 16 ]. We prove Conjecture |${{\mathcal{O}}}$| for Fano complete intersections in projective spaces. The main tools to compute relevant genus-zero Gromov–Witten invariants with primitive insertions are the genus-one topological recursion relation and Zinger's standard-versus-reduced formula. We also prove a related conjecture of Galkin [ 15 ] for projective Fano complete intersections. [ABSTRACT FROM AUTHOR]

For any natural numbers k and l < k, it is defined the action of the algebraic torus (C∗)k on CPN-1, where N = ..., which is given as the l-symmetric power representation of (C∗)k in (C∗)N and the standard action of (C∗)N on CPN-1. It is well known question about description of toric manifolds arising from this action. In this note we solve this problem for k = 2n and l = n, that is we describe the orbits of this action and their closures, that is the corresponding toric manifolds. In this context we also discuss the (C*)2n-action on the complex Grassmann manifolds G2n,n by the Pl¨ucker embedding G2n,n → CPN-1, where N = .... The explicit expressions of torus orbit closures we obtain can be further used in description of singularities of toric varieties. [ABSTRACT FROM AUTHOR]

Let P λ Σ n be the Ehrhart polynomial associated to an integral multiple λ of the standard simplex Σ n ⊂ ℝ n . In this paper, we prove that if (M , L) is an n -dimensional polarized toric manifold with associated Delzant polytope Δ and Ehrhart polynomial P Δ such that P Δ = P λ Σ n , for some λ ∈ ℤ + , then (M , L) ≅ (ℂ P n , O (λ)) (where O (1) is the hyperplane bundle on ℂ P n ) in the following three cases: (1) arbitrary n and λ = 1 , (2) n = 2 and λ = 3 and (3) λ = n + 1 under the assumption that the polarization L is asymptotically Chow semistable. [ABSTRACT FROM AUTHOR]

Motivated by their role in M‐theory, F‐theory, and S‐theory compactifications, all possible complete intersections Calabi‐Yau five‐folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the 3909 product manifolds among those, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non‐product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. The distributions of the invariants are presented, and a comparison with the lower‐dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier, and regressor (both fully connected and convolutional) neural networks. h1, 1 can be learnt very efficiently, with very high R2 score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For h1,4,h2,3,η$h^{1,4},h^{2,3}, \eta$, very high R2 scores are also found, but the accuracy is lower, due to the large ranges of possible values. [ABSTRACT FROM AUTHOR]

Let X = X 1 ∪ ⋯ ∪ X s ⊂ P n , n ≥ 4 , be a general union of smooth non-special curves with X i of degree d i and genus g i and d i ≥ max { 2 g i - 1 , g i + n } if g i > 0 . We prove that X has maximal rank, i.e., for any t ∈ N either h 0 (I X (t)) = 0 or h 1 (I X (t)) = 0 if it is so in a few explicit cases in P 4 . We also prove an unconditional weaker result, maximal rank up to a positive integer δ n . [ABSTRACT FROM AUTHOR]

We construct germs of complex manifolds of dimension m$m$ along projective submanifolds of dimension n$n$ with ample normal bundle and without nonconstant meromorphic functions whenever m⩾2n$m \geqslant 2n$. We also show that our methods do not allow the construction of similar examples when m<2n$m < 2n$ by establishing an algebraicity criterion for foliations on projective spaces that generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence. [ABSTRACT FROM AUTHOR]

We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an n - 1 -dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension 2 n + 1 which are n - 1 -connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four seven-dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein. [ABSTRACT FROM AUTHOR]

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters. [ABSTRACT FROM AUTHOR]

The study of Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian G 2 (ℂ m + 2) is first investigated. It is proved that there exists a shrinking Ricci-Bourguignon soliton on a Hopf real hypersurface M in G 2 (ℂ m + 2) by using pseudo-anticommuting Ricci tensor. Moreover, we have proved that there does not exist a nontrivial gradient Ricci-Bourguignon soliton (M , ξ , ν , ρ , γ , g) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian G 2 (ℂ m + 2). Among the class of contact hypersurface in G 2 (ℂ m + 2) , we also prove that there does not exist a nontrivial gradient Ricci-Bourguignon in G 2 (ℂ m + 2) over the totally geodesic and totally real quaternionic projective space ℍ P n in G 2 (ℂ m + 2) , m = 2 n. [ABSTRACT FROM AUTHOR]

Let X r , s n denote the blow-up of ℙ n along r general lines and s general points. In this paper, we focus on l -very ample line bundles on X 0 , s n and investigate their Seshadri constants with some restrictions on s. Additionally, we compute the nef cone of X r , 0 3 for r ≤ 6 and study the Seshadri constants of some ample line bundles on it. We also examine the Seshadri constants of some ample line bundles on X r , 0 4 (r ≤ 7) and X r , 0 5 (r ≤ 5). [ABSTRACT FROM AUTHOR]

Difference schemes that approximate dynamic systems are considered discrete models of the same phenomena that are described by continuous dynamic systems. Difference schemes with t-symmetry and midpoint and trapezoid schemes are considered. It is shown that these schemes are dual to each other, and, from this fact, we derive theorems on the inheritance of quadratic integrals by these schemes (Cooper's theorem and its dual theorem on the trapezoidal scheme). Using examples of nonlinear oscillators, it is shown that these schemes poses challenges for theoretical research and practical application due to the problem of extra roots: these schemes do not allow one to unambiguously determine the final values from the initial values and vice versa. Therefore, we consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible schemes with t-symmetry can be easily constructed for any dynamical system with a quadratic right-hand side. As an example of such a dynamic system, a top fixed at its center of gravity is considered in detail. In this case, the discrete theory repeats the continuous theory completely: (1) the points of the approximate solution lie on some elliptic curve, which at Δ t → 0 turns into an integral curve; (2) the difference scheme can be represented using quadrature; and (3) the approximate solution can be represented using an elliptic function of a discrete argument. The last section considers the general case. The integral curves are replaced with closures of the orbits of the corresponding Cremona transformation as sets in the projective space over R. The problem of the dimension of this set is discussed. [ABSTRACT FROM AUTHOR]

We classify globally generated vector bundles on a general hyperplane section of P² × P² embedded by the Segre embedding, considering small first Chern classes c1 = (1, 1) and c1 = (2, 1). [ABSTRACT FROM AUTHOR]

We study two quite different types of Terracini loci for the order d-Veronese embedding of an n-dimensional projective space: the minimal one and the primitive one (defined in this paper). The main result is that if n = 4 , d ≥ 19 and x ≤ 2 d , no subset with x points is a minimal Terracini set. We give examples that show that the result is sharp. We raise several open questions. [ABSTRACT FROM AUTHOR]

We construct two new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the three-dimensional complex projective space. In the first series the sheaves have an even first Chern class, and in the second series they have an odd one, while the second and third Chern classes can be expressed as polynomials of a special form in three integer variables. We prove the uniqueness of components in these series for the Chern classes given by those polynomials. [ABSTRACT FROM AUTHOR]

A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested on some four-dimensional manifolds such as the unit sphere S4, the complex projective space CP2, and the product manifold S2\times S2. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians. [ABSTRACT FROM AUTHOR]

Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL$\mathrm{PGL}$ parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn$\iota \colon \mathbb {P}^r\hookrightarrow \mathbb {P}^n$ and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid. [ABSTRACT FROM AUTHOR]

In Neri et al. (2022) [20] , the authors extended the family of 2-dimensional F q 2 t -linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t ≥ 5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t ≥ 5. Moreover, we prove that in the case when t = 4 , the linear sets of the projective line PG (1 , q 8) ensuing from codes in the relevant family, are not equivalent to any one known so far. [ABSTRACT FROM AUTHOR]

Locally convex (or nondegenerate) curves in the sphere S n (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order n + 1. Taking Frenet frames allows us to translate such curves γ into corresponding curves Γ in the flag space, the orthogonal group SO n + 1 or its double cover Spin n + 1. Determining the homotopy type of the space of such closed curves or, more generally, of spaces of such curves with prescribed initial and final jets appears to be a hard problem, which has been solved for n = 2 but otherwise remains open. This paper is a step towards solving the problem for larger values of n. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro regarding the behavior of fundamental systems of solutions to linear ordinary differential equations. We define the itinerary of a locally convex curve Γ : [ 0 , 1 ] → Spin n + 1 as a (finite) word w in the alphabet S n + 1 \ { e } of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of Spin n + 1 pierced by Γ (t) as t ranges from 0 to 1. We prove that, for each word w, the subspace of curves of itinerary w is an embedded contractible (globally collared topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We study both a space of curves with minimum regularity hypotheses, where only topological transversality applies, and spaces of sufficiently regular curves, where transversality has the usual meaning. In both cases we also study the adjacency relation between strata. This is an important step in the construction of CW cell complexes mapped into the original space of curves by weak homotopy equivalences. Our stratification is not as nice as might be desired, lacking for instance the Whitney property. Somewhat surprisingly, the differentiability class of the curves affects some properties of the stratification. The necessary ingredients for the construction of a dual CW complex are proved. [ABSTRACT FROM AUTHOR]