Tangent Cone Research Articles

Local quasi-isometries and tangent cones of definable germs

Abstract In this paper, we introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a particular case of local quasi-isometric equivalence, we obtain Sampaio’s tangent cone theorem as a corollary. As an application, we provide a different proof of the theorem by Fernandes-Sampaio, which states that the tangent cone of a Lipschitz normally embedded germ is also Lipschitz normally embedded.

Gorensteinness for normal tangent cones of elliptic ideals

Let A be a two-dimensional excellent normal Gorenstein local domain. In this paper, we characterize elliptic ideals I⊂A for its normal tangent cone G‾(I) to be Gorenstein. Moreover, we classify all those ideals in a Gorenstein elliptic singularity in the characteristic zero case.

Betti Numbers of the Tangent Cones of Monomial Space Curves

Betti Numbers of the Tangent Cones of Monomial Space Curves

Optimal coordinates for Ricci-flat conifolds

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold (M, g) which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold (Mn,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(M^n,g)$$\\end{document} is of order n and thereby close a small gap in a paper by Cheeger and Tian.

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: “The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing.” We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.

Clarke’s Tangent Cones, Subgradients, Optimality Conditions, and the Lipschitzness at Infinity

Clarke’s Tangent Cones, Subgradients, Optimality Conditions, and the Lipschitzness at Infinity

Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves

Let C be a Gorenstein noncomplete intersection monomial curve in the 4-dimensional affine space with defining ideal I(C). In this article, we use the minimal generating set of I(C) to give a criterion for determining whether the tangent cone of C is Cohen-Macaulay. We also show that if the tangent cone of C is Cohen-Macaulay, then the minimal number of generators of the ideal I(C)⁎ is either 5 or an even integer of the form 2d+2, for a suitable integer d. Additionally, we provide a family of Gorenstein noncomplete intersection monomial curves C whose tangent cone is Cohen-Macaulay and the minimal number of generators of I(C)⁎ is large.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Identification of Real and Complex Solution Varieties and Their Singularities Defined by Loop Constraints of Linkages Using the Kinematic Tangent Cone

Abstract The configuration space (c-space) of a mechanism is the real-solution variety of a set of loop closure constraints, which is therefore the chief object in kinematic analysis. Singularities of this variety (referred to as c-space singularities) are singular configurations of the mechanism. In addition, a mechanism may exhibit other kinematic singularities that are not visible from the differential geometry of the c-space (referred to as hidden singularities). Such situations were analyzed by investigating the local geometry of the c-space and its corank stratification. It has been shown recently that hidden singularities and shakiness can be attributed to the fact that complex solution branches intersect with the c-space, i.e., with real-solution branches. This paper employs the kinematic tangent cone to identify local solution branches. While the kinematic tangent cone is an established generally applicable concept, which gives rise to a computational (numeric and symbolic) algorithm, it has yet only been applied to analyzing the real-solution set. Application of the method is shown for several examples. Further, the algebraic aspects are briefly elaborated.

Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

AbstractFor any $$\Lambda >0$$ Λ > 0 , let $$\mathcal {M}_{n,\Lambda }$$ M n , Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space $$\mathbb {R}^{n+m}$$ R n + m with uniformly bounded 2-dilation $$\Lambda $$ Λ of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone C of $$M\in \mathcal {M}_{n,\Lambda }$$ M ∈ M n , Λ at infinity has multiplicity one. This enables us to get a Neumann–Poincaré inequality on stationary indecomposable components of C. A corollary is a Liouville theorem for M. For small $$\Lambda >1$$ Λ > 1 (we can take any $$\Lambda <\sqrt{2}$$ Λ < 2 ), we prove that (i) for $$n\le 7$$ n ≤ 7 , M is flat; (ii) for $$n>8$$ n > 8 and a non-flat M, any tangent cone of M at infinity is a multiplicity one quasi-cylindrical minimal cone in $$\mathbb {R}^{n+m}$$ R n + m whose singular set has dimension $$\le n-7$$ ≤ n - 7 .

On the local structure of the Brill–Noether locus of locally free sheaves on a smooth variety

We study the functor \operatorname{Def}_{E}^{k} of infinitesimal deformations of a locally free sheaf E of \mathcal{O}_{X} -modules on a smooth variety X , such that at least k independent sections lift to the deformed sheaf. We deduce some information on the k th Brill–Noether locus of E , such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor \operatorname{Def}_{E}^{k} and the smoothness of some well-known deformation functors and their associated moduli spaces. As a tool for the investigation of \operatorname{Def}_{E}^{k} , we study infinitesimal deformations of the pairs (E,U) , where U is a linear subspace of sections of E . We generalise many classical results concerning the moduli space of coherent systems to the case where E has any rank and X any dimension. This includes a description of its tangent space and the link between smoothness and the injectivity of the Petri map.

Rectifiability of Flat Singular Points for Area-Minimizing mod 2Q Hypercurrents

Abstract Consider an $ m $-dimensional area minimizing mod$ (2Q) $ current $ T $, with $ Q\in {\mathbb {N}} $, inside a sufficiently regular Riemannian manifold of dimension $ m + 1 $. We show that the set of singular density-$ Q $ points with a flat tangent cone is $ (m-2) $-rectifiable. This complements the thorough structural analysis of the singularities of area-minimizing hypersurfaces modulo $ p $ that has been completed in the series of works of De Lellis–Hirsch–Marchese–Stuvard and De Lellis–Hirsch–Marchese–Stuvard–Spolaor, and the work of Minter–Wickramasekera.

Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections

We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0$$\\end{document} that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by C×V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes V_0$$\\end{document}. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, by rewriting the cardinality constraint as a constraint requiring that any feasible point must lie in the union of certain subspaces, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.

Local structure of convex surfaces

A point on the surface of a convex body and a supporting plane to the body at this point are under consideration. A plane parallel to this supporting plane and cutting off part of the surface is drawn. The limiting behaviour of the cut-off part of the surface as the cutting plane approaches the point in question is investigated. More precisely, the limiting behavior of the appropriately normalized surface area measure in $S^2$ generated by this part of the surface is studied. The cases when the point is regular and singular (a conical or a ridge point) are considered. The supporting plane can be positioned in different ways with respect to the tangent cone at the point: its intersection with the cone can be a vertex, a line (if a ridge point is considered), a plane angle (which can degenerate into a ray or a half-plane), or a plane (if the point is regular and, correspondingly, the cone degenerates into a half-space). In the case when the intersection is a ray, the plane can be tangent (in a one- or two-sided manner) or not tangent to the cone. It turns out that the limiting behaviour of the measure can be different. In the case when the intersection of the supporting plane and the cone is a vertex or in the case of a (one- or two-sided) tangency, the weak limit always exists and is uniquely determined by the plane and the cone. In the case when the intersection is a line or a ray with no tangency, there may be no limit at all. In this case all possible weak partial limits are characterized. Bibliography: 13 titles.

Characterizations of the set of integer points in an integral bisubmodular polyhedron

In this note, we provide two characterizations of the set of integer points in an integral bisubmodular polyhedron. Our characterizations do not require the assumption that a given set satisfies the hole-freeness, i.e., the set of integer points in its convex hull coincides with the original set. One is a natural multiset generalization of the exchange axiom of a delta-matroid, and the other comes from the notion of the tangent cone of an integral bisubmodular polyhedron.

Linearization of differential inclusions

In this paper we extend the approach of Dubovickiĭ and Miljutin for linearization of the dynamics of smooth control systems to a non-smooth setting. We consider dynamics governed by a differential inclusion and we study the Clarke tangent cone to the set of all admissible trajectories starting from a fixed point. Our approach is based on the classical Filippov’s theorem and on the important property “subtransversality” of two closed sets.

Construction of Free Curves by Adding Lines to a Given Curve

In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C, lines joining the singularities of the curve C, or lines in the tangent cone of some singularities of C. These lines L have in common that the intersection C∩L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\cap L$$\\end{document} consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by Beorchia and Miró-Roig in arXiv:2303.04665

Generalized Lagrange Theorem

The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of the derivative by the so called bisequential tangent cone. We first prove Rolle and Lagrange type results and then we turn to comparing this cone with the Clarke subdifferential in the case of a Lipschitz function. We also investigate an approach using normal cones.

Mechanism singularities and shakiness from an algebraic viewpoint

Kinematic singularities are configurations where the configuration space (c-space) V is not smooth or configurations where the rank of the constraint Jacobian drops while the c-space is locally smooth. Both can be analyzed by means of a higher-order analysis, for which the kinematic tangent cone provides a conceptual framework. It is desirable to complement such an analysis with more insight into the origin of specific singular phenomena. Such additional insight is obtained by investigating the kinematics from an algebraic viewpoint. Points of interest are then the singularities of a variety, referred to as algebraic singularities. Algebraic singularities are reflected as kinematic singularities or as shakiness. It is shown that these algebraic singularities can be attributed to properties of the ideal generated by the loop constraints — the constraint ideal. Particular situations arise depending on whether this constraint ideal is radical. If it is not radical, a singular branch is locally defined by a primary ideal. If it is radical, it must be distinguished whether real and complex dimensions of intersecting branches are different. Real intersections indicate kinematic singularities; in all other cases, the mechanism may be in a singularity or may be shaky. Various examples are presented.