IA Scholar Query: Common Transversals and Tangents to Two Lines and Two Quadrics in P.
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Internet Archive Scholar query results feedeninfo@archive.orgThu, 22 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On certain quantifications of Gromov's non-squeezing theorem
https://scholar.archive.org/work/wmgimfyqeng5nk6cl7mdvm3fqy
Let R>1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder 𝔻^2 ×ℝ^2. By Gromov's non-squeezing theorem, E must be non-empty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R ≤√(2). In an appendix by Joé Brendel, it is shown that the lower bound is optimal for R < √(3). We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.Kevin Sackel, Antoine Song, Umut Varolgunes, Jonathan J. Zhuwork_wmgimfyqeng5nk6cl7mdvm3fqyThu, 22 Sep 2022 00:00:00 GMTToric polar maps and characteristic classes
https://scholar.archive.org/work/tdcph5w67rb7vhb3q7k6uqsx34
Given a hypersurface in the complex projective space, we prove that the degree of its toric polar map is given by the signed topological Euler characteristic of a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes. In addition, we prove that if the hypersurface is in general position or is nondegenerate with respect to its Newton polytope, then the coefficients of the Chern-Schwartz-MacPherson class of the distinguished open set agree, up to sign, with the multidegrees of the toric polar map. In the latter case, we also recover the multidegrees from mixed volumes. For plane curves, a precise formula for the degree of the toric polar map is obtained in terms of local invariants. Finally, we construct families, in arbitrary dimension, of irreducible hypersurfaces whose toric polar map is birational.Thiago Fassarella, Nivaldo Medeiros, Rodrigo Salomãowork_tdcph5w67rb7vhb3q7k6uqsx34Fri, 16 Sep 2022 00:00:00 GMTDegeneration of Hodge structures on I-surfaces
https://scholar.archive.org/work/ophp55k2ffb2bdaujjl4f5m3di
Work of Green, Griffiths, Laza, and Robles suggests that the moduli space of (smoothable) stable surfaces should admit a natural stratification defined via Hodge theoretic data. In the case of stable surfaces with K_X^2 = 1 and χ(X) = 3 we compute the Hodge type of all examples known to us and show that all predicted degenerations are geometrically realised.Stephen Coughlan, Marco Franciosi, Rita Pardini, Sönke Rollenskework_ophp55k2ffb2bdaujjl4f5m3diThu, 15 Sep 2022 00:00:00 GMTFano's Last Fano
https://scholar.archive.org/work/hfojtaj5rngajkeoajmn3g6yo4
In 1949 Fano published his last paper on 3-folds with canonical sectional curves. There he constructed and described a 3-fold of the type X^22_3 in ℙ^13 with canonical curve section, which we like to call Fano's last Fano. We report on Fano's construction, providing various (in our opinion missing) proofs, in modern language and trying to use results and techniques available at that time. Then we construct Fano's with modern tools, in particular via the Hilbert scheme of zero cycles on a rational surface; as a consequence we easily point out the corresponding example in the Mori-Mukai classification.Marco Andreatta, Roberto Pignatelliwork_hfojtaj5rngajkeoajmn3g6yo4Thu, 15 Sep 2022 00:00:00 GMTSmall resolutions of special three-dimensional varieties
https://scholar.archive.org/work/xhvwn2dszbb4hf2tnezq3frq4u
This is an English translation of Werner's 1987 dissertation "Kleine Aufl\"osungen spezieller dreidimensionaler Variet\"aten," originally published as Bonner Mathematische Schriften No. 186.Jürgen Werner, Simon Venter, Nicolas Addingtonwork_xhvwn2dszbb4hf2tnezq3frq4uTue, 13 Sep 2022 00:00:00 GMTGeometric Manin's Conjecture for Fano 3-Folds
https://scholar.archive.org/work/hwms6q6wfjhrnclguhkdc2t6fu
We classify families of irreducible, nef rational curves on general members of all 88 families of smooth Fano threefolds of Picard rank at least two. This proves Geometric Manin's Conjecture for general members of all 88 families, and for arbitrary members of 75 families.Andrew Burke, Eric Jovinellywork_hwms6q6wfjhrnclguhkdc2t6fuMon, 12 Sep 2022 00:00:00 GMTConfigurations of points in projective space and their projections
https://scholar.archive.org/work/vjfv3ep6qfazzkxvzjslejzmay
We call a set of points Z⊂ℙ^3_ℂ an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had ab=12, 16, 20, 24, 30, 36, 42, 48, 54 or 60. Here, for any 4 ≤ a ≤ b, we construct nongrid nondegenerate (a,b)-geproci sets in a systematic way. We also show that the only such example with a=3 is a (3,4)-geproci set coming from the D_4 root system, and we describe the D_4 configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of d-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg, Justyna Szpondwork_vjfv3ep6qfazzkxvzjslejzmaySun, 11 Sep 2022 00:00:00 GMTA unified Casson-Lin invariant for the real forms of SL(2)
https://scholar.archive.org/work/olhr6trhw5gmldua3ayobjspxa
We introduce a unified framework for counting representations of knot groups into SU(2) and SL(2, ℝ). For a knot K in the 3-sphere, Lin and others showed that a Casson-style count of SU(2) representations with fixed meridional holonomy recovers the signature function of K. For knots whose complement contains no closed essential surface, we show there is an analogous count for SL(2, ℝ) representations. We then prove the SL(2, ℝ) count is determined by the SU(2) count and a single integer h(K), allowing us to show the existence of various SL(2, ℝ) representations using only elementary topological hypotheses. Combined with the translation extension locus of Culler-Dunfield, we use this to prove left-orderability of many 3-manifold groups obtained by cyclic branched covers and Dehn fillings on broad classes of knots. We give further applications to the existence of real parabolic representations, including a generalization of the Riley Conjecture (proved by Gordon) to alternating knots. These invariants exhibit some intriguing patterns that deserve explanation, and we include many open questions. The close connection between SU(2) and SL(2, ℝ) comes from viewing their representations as the real points of the appropriate SL(2, ℂ) character variety. While such real loci are typically highly singular at the reducible characters that are common to both SU(2) and SL(2, ℝ), in the relevant situations, we show how to resolve these real algebraic sets into smooth manifolds. We construct these resolutions using the geometric transition S^2 →𝔼^2 →ℍ^2, studied from the perspective of projective geometry, and they allow us to pass between Casson-Lin counts of SU(2) and SL(2, ℝ) representations unimpeded.Nathan M. Dunfield, Jacob Rasmussenwork_olhr6trhw5gmldua3ayobjspxaWed, 07 Sep 2022 00:00:00 GMTEnumerative geometry meets statistics, combinatorics and topology
https://scholar.archive.org/work/mymjdcj7srhfdifnghnhsiudxm
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of Lorentzian polynomials. The main concept joining the mentioned fields is a tensor, or more precisely a linear space of matrices.Mateusz Michałekwork_mymjdcj7srhfdifnghnhsiudxmTue, 06 Sep 2022 00:00:00 GMTConvex hulls of surfaces in fourspace
https://scholar.archive.org/work/txk2fckbz5di7l46bftkkbb2am
This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese, Del Pezzo and Bordiga surfaces.Chiara Meroni, Kristian Ranestad, Rainer Sinnwork_txk2fckbz5di7l46bftkkbb2amFri, 02 Sep 2022 00:00:00 GMTFlexible domains for minimal surfaces in Euclidean spaces
https://scholar.archive.org/work/ede7flvdarevhj5zme4iqn5be4
In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces ℝ^n for n≥ 3 in terms of minimal surfaces which they contain. A domain Ω in ℝ^n is said to be flexible if every conformal minimal immersion U→Ω from a Runge domain U in an open conformal surface M can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion M→Ω. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.Barbara Drinovec Drnovsek, Franc Forstnericwork_ede7flvdarevhj5zme4iqn5be4Thu, 01 Sep 2022 00:00:00 GMTA new family of 3-dimensional space-like Zoll manifolds
https://scholar.archive.org/work/n75enrrkcnf37c4cy5srm5y5iu
We provide a new family of indefinite Einstein-Weyl structures of signature (++-) on a 3-manifold, which are real analytic and space-like Zoll. They are obtained by using the minitwistor correspondence. The minitwistor spaces we use are Segre quartic surfaces of a particular type. They admit a ℂ^*-action and form a 1-dimensional moduli space. Correspondingly, the present Einstein-Weyl spaces admit a circle action and constitute a real 1-dimensional familyNobuhiro Honda, Fuminori Nakatawork_n75enrrkcnf37c4cy5srm5y5iuMon, 29 Aug 2022 00:00:00 GMTModel projective twists and generalised lantern relations
https://scholar.archive.org/work/2rmwp3do2bcjbgnivurrvg3jbq
We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists τ_𝔸ℙ^2∈Symp_ct(T^*𝔸ℙ^2), 𝔸∈{ℝ, ℂ}. In each case, we construct an exact Lefschetz fibration π T^*𝔸ℙ^2→ℂ with three singular fibres, and define a compactly supported symplectomorphism φ∈Symp_ct(T^*𝔸ℙ^2) on the total space. Given two disjoint Lefschetz thimbles Δ_α,Δ_β⊂ T^*𝔸ℙ^2, we compute the Floer cohomology groups HF(φ^k(Δ_α), Δ_β; ℤ/2ℤ) and verify (partially for ℂℙ^2) that φ is indeed isotopic to (a power of) the projective twist in its local model. The constructions we present are governed by generalised lantern relations, which provide an isotopy between the total monodromy of a Lefschetz fibration and a fibred twist along an S^1-fibred coisotropic submanifold of the smooth fibre. We also use these relations to generate non-exact fillings for the contact manifolds (ST^*ℂℙ^2, ξ_std), (ST^*ℝℙ^3,ξ_std), and to study two classes of monotone Lagrangian submanifolds of (T^*ℂℙ^2, dλ_ℂℙ^2).Brunella Charlotte Torricelliwork_2rmwp3do2bcjbgnivurrvg3jbqTue, 23 Aug 2022 00:00:00 GMTUniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents
https://scholar.archive.org/work/ftnmmghjjnghlaoagnqrx6vusy
We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, M\"oller, and Zorich on Lyapunov exponents. Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity. Additionally, we classify the hypergeometric differential equations that satisfy our assumptions. We obtain several multi-parameter families of equations, which include the mirror quintic as well as the six other thin cases of Doran--Morgan and Brav--Thomas.Simion Filipwork_ftnmmghjjnghlaoagnqrx6vusyTue, 23 Aug 2022 00:00:00 GMTExtra-twisted connected sum G_2-manifolds
https://scholar.archive.org/work/bi2vdrqx2rc5dectnpbz5wln24
We present a construction of closed 7-manifolds of holonomy G_2, which generalises Kovalev's twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author arXiv:1505.02734 use this to exhibit examples of closed 7-manifolds with disconnected moduli space of holonomy G_2 metrics.Johannes Nordströmwork_bi2vdrqx2rc5dectnpbz5wln24Thu, 18 Aug 2022 00:00:00 GMTFano 3-folds and classification of constantly curved holomorphic 2-spheres of degree 6 in the complex Grassmannian G(2,5)
https://scholar.archive.org/work/grdsb5f54ngzhacvcrih7ooohu
Up to now the only known example in the literature of constantly curved holomorphic 2-sphere of degree 6 in the complex G(2,5) has been the first associated curve of the Veronese curve of degree 4. By exploring the rich interplay between the Riemann sphere and projectively equivalent Fano 3-folds of index 2 and degree 5, we prove, up to the ambient unitary equivalence, that the moduli space of generic (to be precisely defined) such 2-spheres is semialgebraic of dimension 2. All these 2-spheres are verified to have non-parallel second fundamental form except for the above known example.Quo-Shin Chi, Zhenxiao Xie, Yan Xuwork_grdsb5f54ngzhacvcrih7ooohuWed, 17 Aug 2022 00:00:00 GMTBrill-Noether theory and Green's conjecture for general curves on simple abelian surfaces
https://scholar.archive.org/work/phpgrpuvwzhjddxpts26pkgiyu
In this paper we compute the gonality and the dimension of the Brill-Noether loci W^1_d(C) for curves in a non primitive linear system of a simple abelian surface, adapting vector bundles techniques à la Lazarsfeld originally introduced with K3 surfaces. As a corollary, we obtain general Green's conjecture for curves on abelian surfaces.Federico Morettiwork_phpgrpuvwzhjddxpts26pkgiyuMon, 15 Aug 2022 00:00:00 GMTHomological mirror symmetry at large volume
https://scholar.archive.org/work/sea3wsimnzd2zbzlfmsz32edyy
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove homological mirror symmetry: the category of coherent sheaves on the first space is equivalent to the Fukaya category of the second. Our equivalence intertwines the Viterbo restriction maps for a generalized pair-of-pants cover of the symplectic manifold with the restriction of coherent sheaves for a certain affine cover of the algebraic variety. We deduce a posteriori a local-to-global principle conjectured by Seidel -- certain diagrams of Viterbo restrictions are cartesian -- by passing Zariski descent through our mirror symmetry result.Benjamin Gammage, Vivek Shendework_sea3wsimnzd2zbzlfmsz32edyyFri, 12 Aug 2022 00:00:00 GMTDEM analysis of micromechanics and buffering capacity of superquadric mixture granular materials under impact load
https://scholar.archive.org/work/yw6oaltopfhvvey2zqj35tzwvy
As one of the most common geological disasters, rockfalls seriously threaten the safety of linear projects such as roads, railways, and oil and natural gas pipelines. The rigid protective structures that are used for disaster reduction are easily damaged by the impact of rockfalls, which affects the service life of structures. Consequently, the buffer layer has been introduced to resolve this problem. In this work, numerical simulations were carried out by the discrete element method to study the interaction between falling rocks and the granular medium of a soil cushion layer that is installed on a rigid structure. The falling rock is modeled as a single sphere and the soil cushion layer is modeled as a component composed of a collection under the action of gravity, where the filled particles of the soil cushion layer are based on superquadric spheres generated by the superquadric surface equation. This paper uses three shapes (i.e., spheres, cubes, and cylinders) to mix and match as the soil cushion layer. The buffer performance of different mixed material buffer layers is investigated by analyzing the pressure of the bottom plate. The force chain propagation process is investigated by analyzing the comparison of the force chains of the soil cushion layers with different thickness and different filling particles after being impacted. The energy propagation process was studied by analyzing the evolution of the kinetic energy of the particles after the impact of the soil cushion layer.Hongzhi Qiu, Jintao Yuan, Peifeng Han, Miao Yang, Wenyao Huang, Xu Fang, Yuxin Liwork_yw6oaltopfhvvey2zqj35tzwvyFri, 12 Aug 2022 00:00:00 GMTUsing Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture
https://scholar.archive.org/work/pyijtm3dbvga5hokztaqveqxv4
The task of recognizing an algebraic surface from its apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.Eriola Hoxhaj, Jean Michel Menjanahary, Josef Schichowork_pyijtm3dbvga5hokztaqveqxv4Wed, 10 Aug 2022 00:00:00 GMT