arXiv research

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

A new method clusters intersecting lines using hypergraphs.

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…

Study of monodromy and vanishing cycles for complete intersection curves.

Santaló calculated the measures for all positions of a moving line segment in which it lies inside a fixed circle and intersects this circle in one or two points. From these measures he concluded hitting probabilities for a line segment thrown randomly onto an unbounded lattice of circles. In the present paper these re…

New Calabi-Yau metrics with conical singularities are created near complex lines.

Curve diffusion flow straightens curves with endpoints on intersecting lines.

A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove a global analogue of this result, as follows. Suppose that two …

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is…

The paper evaluates homology for links in a solid torus with special boundary conditions.

Let $\scr A^*=\{l_1,l_2,\cdots,l_n\}$ be a line arrangement in $\Bbb{CP}^2$, i.e., a collection of distinct lines in $\Bbb{CP}^2$. Let $L(\scr A^*)$ be the set of all intersections of elements of $A^*$ partially ordered by $X\leq Y\Leftrightarrow Y\subseteq X$. Let $M(\scr A^*)$ be $\Bbb{CP}^2-\bigcup\scr A^*$ where $\…

Study shows non-spectrality of certain curves and line segments.

For a pair of points in a smooth closed convex planar curve $γ$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Sy…

Let $Y$ be a hyperbolic surface and let $φ$ be a Laplacian eigenfunction having eigenvalue $-1/4-τ^2$ with $τ>0$. Let $N(φ)$ be the set of nodal lines of $φ$. For a fixed analytic curve $γ$ of finite length, we study the number of intersections between $N(φ)$ and $γ$ in terms of $τ$. When $Y$ is compact and $γ$ a geode…

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by consi…

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…

Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.

A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecan…

Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We st…

In this paper we show that to each planar line arrangement defined over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read off from the combinatorics and relative locations of intersections. This leads to an alternate proof of Wajnryb's gen…

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection …

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus $(\C^*)^n$ is a finite group which can be determined using the torsion part of some associated lattice. Applications are given to the study of characteristic varieties of smooth complex algebraic varie…

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-d…

We investigate intersections of geodesic lines in $H^2$ and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let $γ$ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of $γ$ in $H^2$ is shorter then $γ$. However, a simil…

In this paper, we give a fully detailed exposition of computing fundamental groups of complements of line arrangements using the Moishezon-Teicher technique for computing the braid monodromy of a curve and the Van-Kampen theorem which induces a presentation of the fundamental group of the complement from the braid mono…

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture

Unified framework for intersectionally fair AI models using MIO.

It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we pro…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

Let V be a finite dimensional complex vector space and V^* its dual and let X in P(V) be a smooth projective variety of dimension n and degree d at least two. For a generic n-tuple of hyperplanes H_1,...,H_n in P(V^*)^n, the intersection of X with H_1,...,H_n consists of d distinct points. We define the "discriminant o…

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for $J$-holomorphic curves in almost complex $4$-manifolds to higher dimensions. As an applicat…

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

The main object of study in the paper is the distance from a point to a line in the Riemannian manifold associated with the Heston model. We reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals. One of the main ideas in this paper is to use…

We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 P…

We obtain bounds on the least dimension of an affine space that can contain an $n$-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points. This problem is closely related to the generalized vector field problem, non-singular bilinear maps, and the immersion problem for re…

We study the line bundle mean curvature flow on Kähler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of Kähler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian-Yang-Mil…

We construct potentially new manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ and $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ via rational blowdown surgery along certain $4$-valent plumbing graphs. This way all the graph classes from \cite{weighted} have a represen…

Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the relationship between our classification and the second Poisson cohom…

A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let …

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link that is a satellite of the Hopf link. The explicit construction of the complex pla…

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, …

Analyzes Kodaira-Iitaka dimension and multiplicity using intersection theory.

In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…

Following the very recent line of work on the ``generalized min-max'' (GMM) kernel, this study proposes the ``generalized intersection'' (GInt) kernel and the related ``normalized generalized min-max'' (NGMM) kernel. In computer vision, the (histogram) intersection kernel has been popular, and the GInt kernel generaliz…

The paper establishes a Miyaoka-Yau type inequality for hyperplane arrangements in complex projective space.