Tootfinder

A MILP-Based Solution to Multi-Agent Motion Planning and Collision Avoidance in Constrained Environments
Akshay Jaitly, Jack Cline, Siavash Farzan
https://arxiv.org/abs/2506.21982

A MILP-Based Solution to Multi-Agent Motion Planning and Collision Avoidance in Constrained Environments
We propose a mixed-integer linear program (MILP) for multi-agent motion planning that embeds Polytopic Action-based Motion Planning (PAAMP) into a sequence-then-solve pipeline. Region sequences confine each agent to adjacent convex polytopes, while a big-M hyperplane model enforces inter-agent separation. Collision constraints are applied only to agents sharing or neighboring a region, which reduces binary variables exponentially compared with naive formulations. An L1 path-length-plus-accelera…

Quark mass dependence of a QCD critical point and structure of the Columbia plot
Julian Bernhardt, Christian S. Fischer
https://arxiv.org/abs/2507.21680 https://

Quark mass dependence of a QCD critical point and structure of the Columbia plot
We study the quark-mass dependence of the QCD critical point at varying bare up/down quark masses with fixed strange quark mass. We explore the corresponding second-order critical surface in the three-dimensional Columbia plot and study the extension of the associated crossover hyperplane at real and imaginary baryon chemical potential. To this end, we employ a by now well-tested combination of lattice Yang--Mills theory and a (truncated) version of Dyson--Schwinger equations at $N_f=2+1$ quark…

Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding
Ilaria Cardinali, Luca Giuzzi
https://arxiv.org/abs/2506.21309 https://…

Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding
Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $Λ$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $Λ_{1}$ of $Λ$ represented by the pure tensors $x\otimes ξ$ with $x\in V$ and $ξ\in V^*$ such that $ξ(x)=0$. Regarding $Λ_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(Λ_1)$ arising from it. The code $\mathcal{C}(Λ_1)$ i…

Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
Badre Mounda
https://arxiv.org/abs/2507.18999 https://arxiv.org/pdf/2507…

Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $α\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. On the central fibre $\widetilde{X}_0$, the class $α$ specializes to a $\mathbb{Q}$-linear combination of the hyperplane class and the exceptional $(-2)$-curves coming from the blow-ups of the …

Chromatic quasisymmetric functions for signed graphs
Jean-Christophe Aval, Raquel Melgar
https://arxiv.org/abs/2508.20200 https://arxiv.org/pdf/2508.20200

Chromatic quasisymmetric functions for signed graphs
In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this function for a directed graph, that appears to be in $QSym$, the algebra of quasisymmetric functions, which is of great interest in algebraic combinatorics. Our goal is to extend this work to signed graphs, taking into account the perspective of the hyperplane …

Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers
Juan Migliore, Uwe Nagel
https://arxiv.org/abs/2508.12113 https://arxiv.org/pdf/2…

Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers
A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not well understood. A conjecture of Terao says that freeness of $\cA$ depends only on the intersection lattice of $\cA$. Given an arrangement $\cA$, we define the ideal $J_\cA^{top}$ to be the intersection of the codimension 2 primary components of $J_\cA$. This id…

Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding
Ilaria Cardinali, Luca Giuzzi
https://arxiv.org/abs/2507.16694 ht…

Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding
Let $\barΓ$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $σ$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_σ$ of $\barΓ$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[ξ])\in \barΓ$ to the projective point represented by the pure tensor $x^σ\otimes ξ$, with $ξ(x)=0.$ In [I. Car…

Motives of Certain Hyperplane Sections of Milnor Hypersurfaces
Evan Marth
https://arxiv.org/abs/2507.05434 https://arxiv.org/pdf/2507…

Motives of Certain Hyperplane Sections of Milnor Hypersurfaces
We construct a hyperplane section $Y$ of a Milnor hypersurface associated to a semisimple endomorphism $φ$. Exploiting its structure as a hyperplane section of a projective bundle and its natural torus action, we give a motivic decomposition of $Y$, which encodes both the cellular structure of $Y$ and the arithmetic of the eigenvalues of $φ$. This decomposition is proven without using the "nilpotence principle", that is to say there are no "phantoms".

Stability results of the Bishop-Phelps-Bollob\'as property and the generalized AHSP
Thiago Grando, Elisa R. Santos
https://arxiv.org/abs/2506.17762 htt…

Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP
In this paper, we study the Bishop-Phelps-Bollobás property for operators (BPBp for short). To this end, we investigate the generalized approximate hyperplane series property (generalized AHSP for short) for a pair $(X,Y)$ of Banach spaces, which characterizes when $(\ell_1(X),Y)$ has the BPBp. We prove the following results. For $L$ a locally compact Hausdorff space and $\mathcal{A}$ a unital uniform algebra, if either $(X, \mathcal{C}_0(L,Y))$ or $(X, \mathcal{A}^Y)$ has the BPBp, then so do…

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements
Veronika K\"orber, Tobias Schnieders, Jan Stricker, Jasmin Walizadeh
https://arxiv.org/abs/2508.14538

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements
Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Grünbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all We…

Gratulation an Hanna Cairo.
Die 17-jährige Schülerin hat die Vermutung von Mizohata und Takeuchi aus der harmonischen Analysis beantwortet. Sie fand ein Gegenbeispiel.
Paper: https://arxiv.org/abs/2502.06137

Kinematic flow from the flow of cuts
Ross Glew, Andrzej Pokraka
https://arxiv.org/abs/2508.11568 https://arxiv.org/pdf/2508.11568

Kinematic flow from the flow of cuts
The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive ge…

A resolution of the Gaussian hyperplane tessellation conjecture on the sphere
Sjoerd Dirksen, Nigel Q. D. Strachan
https://arxiv.org/abs/2508.05194 https://

A resolution of the Gaussian hyperplane tessellation conjecture on the sphere
We investigate how many hyperplanes with independent standard Gaussian directions one needs to produce a $δ$-uniform tessellation of a subset $S$ of the Euclidean sphere, meaning that for any pair of points in $S$ the fraction of hyperplanes separating them corresponds to their geodesic distance up to an additive error $δ$. It was conjectured that $δ^{-2}w_*(S)^2$ Gaussian random hyperplanes are necessary and sufficient for this purpose, where $w_*(S)$ is the Gaussian complexity of $S$. We f…

Moduli Theory of the $r$-Braid Arrangement
Vance Blankers, Emily Clader, Iva Halacheva, Haggai Liu, Dustin Ross
https://arxiv.org/abs/2506.18205 https://…

Moduli Theory of the $r$-Braid Arrangement
We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space $\overline{\mathcal{M}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space $\overline{\mathcal{M}}^r_{n}$…

On lower bounds for the F-pure threshold of equigenerated ideals
Benjamin Baily
https://arxiv.org/abs/2506.18891 https://arxiv.org/pd…

On lower bounds for the F-pure threshold of equigenerated ideals
Let $k$ be a field of characteristic $p > 0$ and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Additionally, we describe a new relationship between $\mathrm{fpt}(I)$ and $\mathrm{fpt}(I|_H)$, where $H$ is a general hyperplane through the origin. As an application, for degree-$d$ homogeneous polyno…

Classification by Separating Hypersurfaces: An Entropic Approach
Argimiro Arratia, Mahmoud El Daou, Henryk Gzyl
https://arxiv.org/abs/2507.02732 https://…

Classification by Separating Hypersurfaces: An Entropic Approach
We consider the following classification problem: Given a population of individuals characterized by a set of attributes represented as a vector in ${\mathbb R}^N$, the goal is to find a hyperplane in ${\mathbb R}^N$ that separates two sets of points corresponding to two distinct classes. This problem, with a history dating back to the perceptron model, remains central to machine learning. In this paper we propose a novel approach by searching for a vector of parameters in a bounded $N$-dimensi…

Crosslisted article(s) found for math.RT. https://arxiv.org/list/math.RT/new
[1/1]:
- The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups
Ryo Uchiumi

A simultaneous Abels-Margulis-Soifer lemma
Fanny Kassel, Rafael Potrie
https://arxiv.org/abs/2508.08111 https://arxiv.org/pdf/2508.08111

A simultaneous Abels-Margulis-Soifer lemma
The Abels-Margulis-Soifer lemma states that if a semigroup $Γ$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $Γ$ can be multiplied by an element of some fixed finite subset of $Γ$ so that it becomes proximal (i.e. it acts on the corresponding projective space with an attracting fixed point and a repelling projective hyperplane) and even uniformly proximal (i.e. the distance between the attracting fixed point and the repelli…

Mld versus lct near zero
Florin Ambro
https://arxiv.org/abs/2506.18507 https://arxiv.org/pdf/2506.18507

Mld versus lct near zero
We study the equivalence of approaching zero for two invariants of a singularity: the minimal log discrepancy and the log canonical threshold of the general hyperplane section.

A decomposition of Grassmannian associated with a hyperplane arrangement
Houshan Fu, Weikang Liang, Suijie Wang
https://arxiv.org/abs/2506.08539 https://…

A decomposition of Grassmannian associated with a hyperplane arrangement
The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by Gelfand, Goresky, MacPherson and Serganova. Recently, Liang, Wang and Zhao discovered a novel decomposition of the Grassmannian via an essential hyperplane arrangement, which generalizes the first two methods. However, their work was confined to essential hyperpl…

A Generalization of a Classical Geometric Extremum Problem
Petar Kenderov, Oleg Mushkarov, Nikolai Nikolov
https://arxiv.org/abs/2506.07252 https://…

A Generalization of a Classical Geometric Extremum Problem
Let $\partial \,\mathcal{C}$ be the boundary of a compact convex body $\mathcal{C}$ in $\mathbb{R}^n,\, n\geq 2$, and $O$ be an interior point of $\mathcal C$. Every straight line $l$ containing $O$ cuts from $\mathcal{C}$ a segment $[AB]$ with end-points on $\partial \,\mathcal{C}$. It is shown that if $[AB]$ is the shortest such segment, then $\partial \,\mathcal{C}$ is smooth at the points $A$ and $ B$ (i.e. at both of them there is only one supporting hyperplane for $\mathcal{C}$) and, some…

Stability and rigidity results of space-like hypersurface in the Minkowski space
Jianhua Chen, Haiyun Deng, Haiqin Xie, Jiabin Yin
https://arxiv.org/abs/2506.01012

Stability and rigidity results of space-like hypersurface in the Minkowski space
In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs $x_{n+1}=u(x)$ over domain $Ω\subset\mathbb R^n$, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l

Taming Irrationality: An Invariance Principle for the Random Billiard Walk
Ruben Carpenter
https://arxiv.org/abs/2508.12849 https://arxiv.org/pdf/2508.1284…

Taming Irrationality: An Invariance Principle for the Random Billiard Walk
The random billiard walk is a stochastic process $(L_t)_{t\geq 0}$ in which a laser moves through the Coxeter arrangement of an affine Weyl group in $\mathbb{R}^d$, reflecting at each hyperplane with probability $p\in (0, 1)$ and transmitting unchanged otherwise. Defant, Jiradilok, and Mossel introduced this process from the perspective of algebraic combinatorics and established that, for initial directions aligned with the coroot lattice, $L_t/\sqrt{t}$ converges to a centered spherical Gaussi…

Nondegenerate hyperplane covers of the hypercube
Lisa Sauermann, Zixuan Xu
https://arxiv.org/abs/2507.00773 https://arxiv.org/pdf/250…

Nondegenerate hyperplane covers of the hypercube
We consider collections of hyperplanes in $\mathbb{R}^n$ covering all vertices of the $n$-dimensional hypercube $\{0,1\}^n$, which satisfy the following nondegeneracy condition: For every $v\in \{0,1\}^n$ and every $i=1,\dots,n$, we demand that there is a hyperplane $H$ in the collection with $v\in H$ such that the variable $x_i$ appears with a non-zero coefficient in the hyperplane equation describing $H$. We prove that every collection $\mathcal{H}$ of hyperplanes in $\mathbb{R}^n$ covering $…

Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
Archisman Panigrahi, Bitan Roy
https://arxiv.org/abs/2507.23783 https://

Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
Projected branes are constituted by only a small subset of sites of a higher-dimensional crystal, otherwise placed on a hyperplane oriented at an irrational or a rational slope therein, for which the effective Hamiltonian is constructed by systematically integrating out the sites of the parent lattice that fall outside such branes [Commun. Phys. 5, 230 (2022)]. Specifically, when such a brane is constructed from a square lattice, it gives rise to an aperiodic Fibonacci quasi-crystal or its rati…

Large grid subsets without many cospherical points
Zichao Dong, Zijian Xu
https://arxiv.org/abs/2506.18113 https://arxiv.org/pdf/2506…

Large grid subsets without many cospherical points
Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the previously best known lower bound of $n/4$ toward the Erdős--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $Ω\bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $Ω\bigl( n^{\frac{d}…

This https://arxiv.org/abs/2502.12010 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Hyperplane arrangements and the Gauss map of a pencil
We show that the coefficients of the characteristic polynomial of a central hyperplane arrangement $\mathcal A$, coincide with the multidegrees of the Gauss map of a pencil of hypersurfaces naturally associated to $\mathcal A$. As a consequence, we obtain a proof of the Heron-Rota-Welsh conjecture for matroids representable over a field of characteristic zero.

The full automorphism groups of the five symmetric $(15,8,4)$-designs
Mark Pankov, Krzysztof Petelczyc, Mariusz \.Zynel
https://arxiv.org/abs/2506.17216 ht…

The full automorphism groups of the five symmetric $(15,8,4)$-designs
It is clear that the full automorphism group of the $(15,8,4)$-design of points and hyperplane complements of ${\rm PG}(3,2)$ is ${\rm GL}(4,2)$. Using methods of point-line geometries, we determine the full automorphism groups of the remaining four symmetric $(15,8,4)$-designs.

Gr\"unbaum's inequality for probability measures with concavity properties
Matthieu Fradelizi, Dylan Langharst, Jiaqian Liu, Francisco Mar\'in Sola, Shengyu Tang
https://arxiv.org/abs/2507.06759

Grünbaum's inequality for probability measures with concavity properties
A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. Of particular interest is the case of $s$-concave Radon measures. For such measures, we provide a simpler proof of known results and, more importantl…

Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n 1}$
Qing-Ming Cheng, Guoxin Wei
https://arxiv.org/abs/2507.00342 https://

Complete two-sided $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$
In this paper, we study complete $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $δ$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $δ>δ_0(n)$, where $δ_0(3)=1/3$, $δ_0(4)=1/2$ and $δ_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $δ$-stable minimal hypersurface i…

Interpreting the chromatic polynomial coefficients via hyperplane arrangements
Neha Goregaokar
https://arxiv.org/abs/2506.00941 https://

Interpreting the chromatic polynomial coefficients via hyperplane arrangements
A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements o…

Preorders on maximal chains: hyperplane arrangements, Cambrian lattices, and maximal green sequences
Mikhail Gorsky, Nicholas J. Williams
https://arxiv.org/abs/2506.08858

Preorders on maximal chains: hyperplane arrangements, Cambrian lattices, and maximal green sequences
We study preorders on (equivalence classes of) maximal chains in the general context of polygonal lattices endowed with suitably nice edge labellings. We show that, given a quotient of polygonal lattices, such edge labellings descend to the quotient, and that there is an induced order-preserving surjective map on the preordered sets of equivalence classes of maximal chains. Under a natural condition ensuring that the domain is a poset, the map is a contraction of preordered sets. We apply this …

Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings
Fu Houshan
https://arxiv.org/abs/2506.07071

Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings
We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the unsigned coefficients of the characteristic polynomial form a unimodal and log-concave sequence, extending the Rota-Heron-Welsh Conjecture to semimatroids. Furthermore, we present convolution identities for the multiplicative characteristic and Tutte polynom…

Pierce-Birkhoff conjecture is true for splines
Zehua Lai, Lek-Heng Lim
https://arxiv.org/abs/2507.07976 https://arxiv.org/pdf/2507.07…

Pierce-Birkhoff conjecture is true for splines
We prove the Pierce--Birkhoff conjecture for splines, i.e., continuous piecewise polynomials of degree $d$ in $n$ variables on a hyperplane partition of $\mathbb{R}^n$, can be written as a finite lattice combination of polynomials. We will provide a purely existential proof, followed by a more in-depth analysis that yields effective bounds.

This https://arxiv.org/abs/2308.00959 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On Treibich-Verdier curves
We survey some properties of a class of curves lying on certain elliptic ruled surfaces, studied by A. Treibich and J.L. Verdier in connection with elliptic solitons and KP equations. In particular we discuss their Brill-Noether generality, proved by A. Treibich, and we show that they are limits of hyperplane sections of K3 surfaces.

Arithmetic non-generic arrangements
Pragnya Das, Takuya Saito, Simona Settepanella
https://arxiv.org/abs/2506.23124 https://arxiv.org…

Arithmetic non-generic arrangements
A discriminantal hyperplane arrangement B(n,k,A) is constructed from a given (generic) hyperplane arrangement A, which is classified as either very generic or non-very generic depending on the combinatorial structure of B(n,k,A). In particular, A is considered non-very generic if the intersection lattice of B(n,k,A) contains at least one non-very generic intersection -- that is, an intersection that fails to satisfy a specific rank condition established by Athanasiadis in [1]. In this paper, we…

Some general \'etale Weak Lefschetz-type theorems
Sergei I. Arkhipov, Mikhail V. Bondarko
https://arxiv.org/abs/2507.06816 https://

Some general étale Weak Lefschetz-type theorems
We establish new general etale versions of theorems of Barth and Sommese. Respectively, we compute the lower etale cohomology of closed subvarieties of $P^N$ of small codimensions and of their preimages with respect to proper morphisms (that are not necessarily finite; this statement is completely new), and also of the zero loci of sections of ample vector bundles; all these statements are valid over fields of arbitrary characteristics. To obtain these results, we use a new 'fat hyperplane sect…

Salvetti complexes for conditional oriented matroids
Galen Dorpalen-Barry, Dan Dugger, Nicholas Proudfoot
https://arxiv.org/abs/2507.06365 https://

Salvetti complexes for conditional oriented matroids
We give a new proof of the fact that the complement of the complexification of a real hyperplane arrangement is homotopy equivalent to the Salvetti complex of the associated oriented matroid. Our proof involves no choices, is relatively easy to visualize, and generalizes to the setting of conditional oriented matroids.

This https://arxiv.org/abs/2501.13027 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On projective threefolds with two-dimensional space of vanishing cycles
We obtain a complete list of smooth projective threefolds over $\mathbb C$ for which the dimension of the space of vanishing cycles (in $H^2$ of the smooth hyperplane section) equals $2$. We also obtain a complete list of rank 2 very ample vector bundles $E$ on smooth projective surfaces with $c_2(E) = 3$.

Exponents of 2-multiarrangements of three lines over fields of positive characteristic
Shuhei Tsujie, Ryo Uchiumi
https://arxiv.org/abs/2506.07070 https://…

Exponents of 2-multiarrangements of three lines over fields of positive characteristic
Freeness of multiarrangements is extremely important in the study of freeness of hyperplane arrangements. In particular, the exponents of 2-multiarrangement play an important role in freeness of 3-arrangement. However, it is not easy to compute exponents in general. Wakamiko (2007) determined bases and the exponents for multiarrangements of three lines over a field of characteristic zero. In this paper, we study the exponents of multiarrangements of three lines for the case of positive characte…

Notes on the cohomology of partial Hessenberg varieties
Tatsuya Horiguchi, Mikiya Masuda, Takashi Sato, Haozhi Zeng
https://arxiv.org/abs/2507.23259 https://

Notes on the cohomology of partial Hessenberg varieties
Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this …

This https://arxiv.org/abs/2502.02274 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Hyperpolygonal arrangements
In 2024, Bellamy, Craw, Rayan, Schedler, and Weiss introduced a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties which we thus call hyperpolygonal arrangements $\mathcal H_n$. In this note we study these arrangements and investigate their properties systematically. Remarkably the arrangements $\mathcal H_n$ discriminate between essentially all local properties of arrangements. In addition we show that hyperpolygonal a…

This https://arxiv.org/abs/2402.14287 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tropical Fermat-Weber Polytropes
We study the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well-known that a tropical Fermat-Weber point of a given sample is not unique and we show that the set of all possible Fermat-Weber points forms a polytrope. To prove this, we show that the tropical Fermat-Weber is the dual of a minimum-cost flow problem, and that its polytrope is a bounded cell of a tropical hyperplane arrangement given by both max- and min-…

The pure condition for incidence geometries
Daniel Irving Bernstein, Signe Lundqvist
https://arxiv.org/abs/2507.02661 https://arxiv.o…

The pure condition for incidence geometries
The space of \emph{parallel redrawings} of an incidence geometry $(P,H,I)$ with an assigned set of normals is the set of points and hyperplanes in $\mathbb{R}^d$ satisfying the incidences given by $(P,H,I)$, such that the hyperplanes have the assigned normals. In 1989, Whiteley characterized the incidence geometries that have d-dimensional realizations with generic hyperplane normals such that all points and hyperplanes are distinct. However, some incidence geometries can be realized as point…

This https://arxiv.org/abs/2501.03645 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Big convex polytopes or rich hyperplanes
For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in convex position. In this paper, we give the upper and lower bounds for $ES_d(l,n)$.

Characterization of non-singular hyperplanes of $H\left(s,q^2\right)$ in $\mathrm{P G}\left(s, q^2\right)$
Stuti Mohanty, Bikramaditya Sahu
https://arxiv.org/abs/2506.23330

Characterization of non-singular hyperplanes of $H\left(s,q^2\right)$ in $\mathrm{P G}\left(s, q^2\right)$
In this paper, we present a combinatorial characterization of the hyperplanes associated with non-singular hermitian varieties ${H}\left(s, q^2\right)$ in the projective space $\mathrm{PG}\left(s,q^2\right)$ where $s\geq3$ and $q>2$. By analyzing the intersection numbers of hyperplanes with points and co-dimension $2$ subspaces, we establish necessary and sufficient conditions for a hyperplane to be part of the hermitian variety. This approach extends previous characterizations of hermitian var…