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Continuous CM-regularity and generic vanishing Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Debaditya Raychaudhury
We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, O X (1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative
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Duality related with key varieties of ℚ-Fano threefolds constructed from projective bundles Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Hiromichi Takagi
In our previous paper [31], we show that all primeℚ-Fano 3-folds X with only 1/2(1, 1, 1)-singularities in certain 5 classes can be embedded as linear sections into bigger dimensionalℚ-Fano varieties called key varieties; each key variety is constructed from data of the Sarkisov link starting from the blow-up at one 1/2(1, 1, 1)-singularity of X. In this paper, we introduce varieties associated with
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A note on polarized varieties with high nef value Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Zhining Liu
We study the classification problem for polarized varieties with high nef value. We give a complete list of isomorphism classes for normal polarized varieties with high nef value. This generalizes classical work on the smooth case by Fujita, Beltrametti and Sommese. As a consequence we obtain that polarized varieties with slc singularities and high nef value are birationally equivalent to projective
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The balanced superelliptic mapping class groups are generated by three elements Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Genki Omori
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable
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Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Taro Hayashi
Let X be a K3 surface and let g be a non-symplectic involution of X such that the fixed points set contains a curve of genus 8 or more. In this paper, we show that the quotient space X/〈g〉 is determined by the fixed points set and the action of g on rational curves on X.
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New sextics of genus 6 and 10 attaining the Serre bound Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Annamaria Iezzi, Motoko Qiu Kawakita, Marco Timpanella
We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a generalization of Wiman’s sextics [38] and Edge’s sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries
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Anisotropic area-preserving nonlocal flow for closed convex plane curves Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Tianyu Zhao, Yunlong Yang, Yueyue Mao, Jianbo Fang
We consider an anisotropic area-preserving nonlocal flow for closed convex plane curves, which is a generalization of the model introduced by Pan and Yang (J. Differential Equations 266 (2019), 3764–3786) when τ = 1. Under this flow, the evolving curve maintains its convexity and converges to a homothety of a smooth symmetric strictly convex plane curve in the C ∞ sense. The analysis of the asymptotic
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Bach flow of simply connected nilmanifolds Adv. Geom. (IF 0.5) Pub Date : 2024-01-24 Adam Thompson
The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the L 2-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times
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Closed Majorana representations of {3, 4}+-transposition groups Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Alexander A. Ivanov
The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and
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A new axiomatics for masures II Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Auguste Hébert
Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.
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Some remarks on proper actions, proper metric spaces, and buildings Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Linus Kramer
We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.
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A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Günter F. Steinke
Kleinewillinghöfer classified in 1979 automorphism groups of Laguerre planes with respect to linearly transitive subgroups of central automorphisms and obtained a multitude of types. All feasible Kleinewillinghöfer types of 2-dimensional Laguerre planes were completely determined in 2021. In this paper we investigate the Kleinewillinghöfer types of 4-dimensional Laguerre planes with respect to the
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A geographical study ofM‾2P2,4main $\overline{\mathcal{M}}_2\left(\mathbb{P}^2, 4\right)^{\text {main }}$ Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Luca Battistella, Francesca Carocci
We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation of via logarithmic geometry and Gorenstein singularities.
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On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Toru Tsukioka
We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in the product of a projective space and a projective line. We give a classification result for a special case. We determine the nef cones and describe the extremal contractions for related smooth projective varieties.
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Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Rainer Löwen
Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here
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Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35 Adv. Geom. (IF 0.5) Pub Date : 2023-11-07 Harald Löwe
The present paper investigates 16-dimensional compact translation planes with automorphism groups of dimension d between 35 and 37; planes with groups of higher dimensions have been classified by Hähl. We obtain a complete classification for d = 37 (up to isomorphisms). It turns out that these planes have Lenz type V and are already described in a recent paper of Hähl and Meyer [10]. Moreover, we give
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Projective self-dual polygons in higher dimensions Adv. Geom. (IF 0.5) Pub Date : 2023-10-17 Ana Chavez-Caliz
This paper examines the moduli space M m ,n,k of m-self-dual n-gons in ℙ k . We present an explicit construction of self-dual polygons and determine the dimension of M m ,n,k for certain n and m. Additionally, we propose a conjecture that extends Clebsch’s theorem, which states that every pentagon in ℝℙ2 is invariant under the Pentagram map.
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The geometry of discrete L-algebras Adv. Geom. (IF 0.5) Pub Date : 2023-10-17 Wolfgang Rump
The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is
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On the bond polytope Adv. Geom. (IF 0.5) Pub Date : 2023-10-17 Markus Chimani, Martina Juhnke-Kubitzke, Alexander Nover
While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be
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Ehrhart theory of paving and panhandle matroids Adv. Geom. (IF 0.5) Pub Date : 2023-10-14 Derek Hanely, Jeremy L. Martin, Daniel McGinnis, Dane Miyata, George D. Nasr, Andrés R. Vindas-Meléndez, Mei Yin
We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the
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Universal convex covering problems under translations and discrete rotations Adv. Geom. (IF 0.5) Pub Date : 2023-10-13 Mook Kwon Jung, Sang Duk Yoon, Hee-Kap Ahn, Takeshi Tokuyama
We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete
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Helmut Salzmann and his legacy Adv. Geom. (IF 0.5) Pub Date : 2023-10-13 Rainer Löwen
We describe the development of the mathematics of Helmut R. Salzmann (3. 11. 1930 – 8. 3. 2022) and the main difficulties he was facing, documenting his lifelong productivity and his far reaching influence. We include a comprehensive bibliography of his work.
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Exploring tropical differential equations Adv. Geom. (IF 0.5) Pub Date : 2023-10-12 Ethan Cotterill, Cristhian Garay López, Johana Luviano
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean
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A partial compactification of the Bridgeland stability manifold Adv. Geom. (IF 0.5) Pub Date : 2023-10-11 Barbara Bolognese
Bridgeland stability manifolds of Calabi–Yau categories are of noticeable interest both in mathematics and physics. By looking at some of the known examples, a pattern clearly emerges and gives a fairly precise description of how they look like. In particular, they all seem to have missing loci, which tend to correspond to degenerate stability conditions vanishing on spherical objects. Describing such
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Topology of tropical moduli spaces of weighted stable curves in higher genus Adv. Geom. (IF 0.5) Pub Date : 2023-08-11 Siddarth Kannan, Shiyue Li, Stefano Serpente, Claudia He Yun
We study the topology of moduli spaces of weighted stable tropical curves Δg ,w with fixed genus and unit volume. The space Δg ,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space M g ,w of weighted stable curves. When the genus is positive, we show that Δg ,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic
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Cohomogeneity one central Kähler metrics in dimension four Adv. Geom. (IF 0.5) Pub Date : 2023-08-11 Thalia Jeffres, Gideon Maschler, Robert Ream
A Kähler metric is called central if the determinant of its Ricci endomorphism is constant; see [12]. For the case in which this constant is zero, we study on 4-manifolds the existence of complete metrics of this type which have cohomogeneity one for three unimodular 3-dimensional Lie groups: SU(2), the group E(2) of Euclidean plane motions, and a quotient by a discrete subgroup of the Heisenberg group
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Polyhedral compactifications, I Adv. Geom. (IF 0.5) Pub Date : 2023-08-11 Corina Ciobotaru, Linus Kramer, Petra Schwer
In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral
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Tropical Poincaré duality spaces Adv. Geom. (IF 0.5) Pub Date : 2023-08-11 Edvard Aksnes
The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this
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Explicit p-harmonic functions on the real Grassmannians Adv. Geom. (IF 0.5) Pub Date : 2023-07-17 Elsa Ghandour, Sigmundur Gudmundsson
We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functions on the compact real Grassmannians. We also find proper p-harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.
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On automorphisms of semistable G-bundles with decorations Adv. Geom. (IF 0.5) Pub Date : 2023-07-17 Andres Fernandez Herrero
We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar
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Abelian branched covers of rational surfaces Adv. Geom. (IF 0.5) Pub Date : 2023-07-17 Robert Harris, Amey Joshi, B. Doug Park, Mainak Poddar
We study abelian covers of rational surfaces branched over line arrangements. We use these covers to address the geography problem for closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature.
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Isometries of wall-connected twin buildings Adv. Geom. (IF 0.5) Pub Date : 2023-07-17 Sebastian Bischof, Bernhard Mühlherr
We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.
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Pseudoholomorphic curves on the LCS-fication of contact manifolds Adv. Geom. (IF 0.5) Pub Date : 2023-06-02 Yong-Geun Oh, Yasha Savelyev
For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S 1 = M id (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form
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Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids Adv. Geom. (IF 0.5) Pub Date : 2023-05-27 Rikke Eriksen, Markus Kiderlen
The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations
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Characterizations of symplectic polar spaces Adv. Geom. (IF 0.5) Pub Date : 2023-05-27 Ilaria Cardinali, Hans Cuypers, Luca Giuzzi, Antonio Pasini
Apolar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG(V) such that the image ε(𝒮) of 𝒮 by ε is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 𝒮 admits different (non-isomorphic) embeddings,
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Fano fourfolds having a prime divisor of Picard number 1 Adv. Geom. (IF 0.5) Pub Date : 2023-03-20 Saverio Andrea Secci
We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ1-bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show
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Open books and embeddings of smooth and contact manifolds Adv. Geom. (IF 0.5) Pub Date : 2023-03-20 Arijit Nath, Kuldeep Saha
We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n − k . We then prove that every closed
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Chow groups of Gushel–Mukai fivefolds Adv. Geom. (IF 0.5) Pub Date : 2023-03-20 Lin Zhou
We compute the Chow groups of smooth Gushel–Mukai varieties of dimension 5.
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Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi–Yau varieties Adv. Geom. (IF 0.5) Pub Date : 2023-03-20 Ugo Bruzzo, Armando Capasso
We consider Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced in [5; 4], and show that they have Jordan-Hölder filtrations whose quotients are stable, locally free and H-nflat. This is applied to show that curve semistable Higgs bundles on simply connected Calabi–Yau varieties have vanishing discriminant.
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Shellable tilings on relative simplicial complexes and their h-vectors Adv. Geom. (IF 0.5) Pub Date : 2023-02-01 Jean-Yves Welschinger
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on
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On projections of free semialgebraic sets Adv. Geom. (IF 0.5) Pub Date : 2023-02-01 Tom Drescher, Tim Netzer, Andreas Thom
We examine to what extent a projection theorem is possible in the non-commutative (free) setting. We first review and extend some results that count against a full free projection theorem. We then obtain a weak version of the projection theorem: projections along linear and separated variables yield semialgebraically parametrised free semi-algebraic sets.
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Ulrich trichotomy on del Pezzo surfaces Adv. Geom. (IF 0.5) Pub Date : 2023-01-17 Emre Coskun, Ozhan Genc
We use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization.
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Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two Adv. Geom. (IF 0.5) Pub Date : 2023-01-16 Dae-Won Lee
In this paper, we give a complete classification of full exceptional collections, up to cyclic permutations, normalizations and mutations, on smooth toric Fano threefolds and fourfolds with Picard rank two. For such varieties, we find all the exceptional collections of maximal length and show that they are in fact full. This gives a partial answer to a conjecture in [29] and [32]. Moreover, such full
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Boomerang uniformity of power permutations and algebraic curves over 𝔽2 n Adv. Geom. (IF 0.5) Pub Date : 2023-01-16 Sihem Mesnager, Ferruh Özbudak
We obtain the Boomerang Connectivity Table of power permutations F ( x ) = x 2 m − 1 of F 2 n $F(x)={{x}^{{{2}^{m}}-1}}\text{ }\!\!~\!\!\text{ of }\!\!~\!\!\text{ }{{\mathbb{F}}_{{{2}^{n}}}}$ with m ∈ { 3 , n − 1 2 , n + 1 2 , n − 2 } . $\left\{ 3,\frac{n-1}{2},\frac{n+1}{2},n-2 \right\}.$ In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F ( x ) at b ∈ F 2 n ∖ F
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Erratum "Tropical superelliptic curves" Adv. Geom. (IF 0.5) Pub Date : 2023-01-16 Madeline Brandt, Paul Alexander Helminck
We correct two errors in Tropical superelliptic curves published in Advances in Geometry on October 8th, 2020. These corrections do not change the main results of the paper.
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PL Morse theory in low dimensions Adv. Geom. (IF 0.5) Pub Date : 2023-01-14 Romain Grunert, Wolfgang Kühnel, Günter Rote
We discuss a PL analog of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level; it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular, we show that in dimensions d ≤ 4 a homologically
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Hilton–Milner results in projective and affine spaces Adv. Geom. (IF 0.5) Pub Date : 2023-01-14 Jozefien D’haeseleer
In this article, we analyse maximal sets of k-spaces, in PG(n, q) and AG(n, q) with n ≥ 2k + t + 3, that pairwise meet in at least a t-space. It is known that for both PG(n, q) and AG(n, q), the largest example is a t-pencil, i.e. the set of all k-spaces containing a fixed t-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n, q) and AG(n, q).
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The Malgrange–Galois groupoid of the Painlevé VI equation with parameters Adv. Geom. (IF 0.5) Pub Date : 2022-07-18 David Blázquez-Sanz, Guy Casale, Juan Sebastián Díaz Arboleda
The Malgrange–Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange–Galois groupoid of the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations
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Bridgeland stability conditions on surfaces with curves of negative self-intersection Adv. Geom. (IF 0.5) Pub Date : 2022-07-18 Rebecca Tramel, Bingyu Xia
Let X be a smooth complex projective variety. In 2002, Bridgeland [6] defined a notion of stability for the objects in 𝔇 b (X), the bounded derived category of coherent sheaves on X, which generalised the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on X and the geometry of the variety. We construct new stability conditions for
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Computation of Dressians by dimensional reduction Adv. Geom. (IF 0.5) Pub Date : 2022-07-18 Madeline Brandt, David E. Speyer
We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.
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A note on large Kakeya sets Adv. Geom. (IF 0.5) Pub Date : 2022-07-15 Maarten De Boeck, Geertrui Van de Voorde
A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q 2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q 2 – q + q contain a large knot, and we obtain a sharp result for planes
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Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers Adv. Geom. (IF 0.5) Pub Date : 2022-07-02 Hara Charalambous, Kostas Karagiannis, Sotiris Karanikolopoulos, Aristides Kontogeorgis
We present a necessary and sufficient condition for a maximal curve, defined over the algebraic closure of a finite field, to be realised as an HKG-cover. We use an approach via pole numbers in a rational point of the curve. For this class of curves, we compute their Weierstrass semigroup as well as the jumps of their higher ramification filtrations at this point, the unique ramification point of the
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Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities Adv. Geom. (IF 0.5) Pub Date : 2022-07-02 Daniel Naie
It is shown that a smooth surface lying on a nodal quartic hypersurface in ℙ4 is either regular or an elliptic conic bundle of degree 8. Furthermore, the latter configuration is shown to exist.
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Explicit Nikulin configurations on Kummer surfaces Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 Xavier Roulleau, Alessandra Sarti
A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the 16 curves. In the paper [16], we constructed explicitly non-isomorphic Kummer structures on some Kummer surfaces
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A few more extensions of Putinar’s Positivstellensatz to non-compact sets Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 Paula Escorcielo, Daniel Perrucci
We extend previous results about Putinar’s Positivstellensatz for cylinders of type S × ℝ to sets of type S × ℝ r in some special cases, taking into account r and the degree of the polynomial with respect to the variables moving in ℝ r (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials
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Positive semigroups in lattices and totally real number fields Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 Lenny Fukshansky, Siki Wang
Let L be a full-rank lattice in ℝ d and write L + for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L +. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L +, and we investigate the
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A characterization of centrally symmetric convex bodies in terms of visual cones Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 E. Morales-Amaya, J. Jerónimo-Castro, D. J. Verdusco Hernández
We prove the following result: Let K be a strictly convex body in the Euclidean space ℝ n , n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊 n–1, such that K is contained in the interior of L. Suppose that, for every x ∈ L, there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric
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Geometric characterisation of subvarieties of 𝓔6(𝕂) related to the ternions and sextonions Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 Anneleen De Schepper
The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔6(𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over 𝕂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties
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On the Jacobian locus in the Prym locus and geodesics Adv. Geom. (IF 0.5) Pub Date : 2022-04-19 Sara Torelli
Let Jg be the Jacobian locus and let P g+1 be the Prym locus, in the moduli space Ag of principally polarized abelian varieties of dimension g, for g ≥ 7. We study the extrinsic geometry of Jg ⊂ P g+1 , under the inclusion provided by the theory of generalized Prym varieties introduced by Beauville. More precisely, we address the problem if certain geodesic curves, defined with respect to the Siegel
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Counting isolated points outside the image of a polynomial map Adv. Geom. (IF 0.5) Pub Date : 2022-04-18 Boulos El Hilany
We consider a generic family of polynomial maps f := (f 1, f 2) : ℂ2 → ℂ2 with given supports of polynomials, and degree deg f := max(deg f 1, deg f 2). We show that the (non-) properness of maps f in this family depends uniquely on the pair of supports, and that the set of isolated points in ℂ2 ∖ f(ℂ2) has a size of at most 6 deg f. This improves an existing upper bound (deg f – 1)2 proven by Jelonek