Search results for "Plane curve"

An axiomatic treatment of ratios in an affine plane

1967

Mond's conjecture for maps between curves

2017

A theorem by D. Mond shows that if f:(C,0)→C2,0 is finite and has has degree one onto its image (Y, 0), then the Ae-codimension is less than or equal to the image Milnor number μI(f), with equality if and only if (Y, 0) is weighted homogeneous. Here we generalize this result to the case of a map germ f:(X,0)→C2,0, where (X, 0) is a plane curve singularity.

A constructive theory of shape

2021

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter $\kappa \in (0,\infty)$. We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a va…

Graphs of stable maps from closed surfaces to the projective plane

2018

Abstract We describe how to attach a weighted graph to each stable map from closed surfaces to projective plane and prove that any weighted graph with non negatively weighted vertices is the graph of some stable map from a closed surface to the projective plane.

Real Line Arrangements and Surfaces with Many Real Nodes

2008

A long standing question is if the maximum number μ(d) of nodes on a surface of degree d in P( ) can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, μ(d) 5 12 d, is provided by Chmutov’s construction from 1992 which gives surfaces whose nodes have non-real coordinates. Using explicit constructions of certain real line arrangements we show that Chmutov’s construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov’s lower bound (i.e., for d = 3, 4, . . . , 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up t…

On complete metric spaces containing the Sierpinski curve

1998

It is proved that a complete metric space topologically contains the Sierpiński universal plane curve if and only if it has a subset with so-called bypass property, i.e. it has a subset K K containing an arc such that for each a ∈ K a\in K and for each open arc A ⊂ K A\subset K with a ∈ A a\in A , there exists an arbitrary small arc in K ∖ { a } K\setminus \{a\} joining the two components of A ∖ { a } A\setminus \{a\} .

A Common Characterization of Finite Projective Spaces and Affine Planes

1981

Let S be a finite linear space for which there is a non-negative integer s such that for any two disjoint lines L, L' of S and any point p outside L and L' there are exactly s lines through p intersecting the two lines L and L'. We prove that one of the following possibilities occurs: (i) S is a generalized projective space, and if the dimension of S is at least 4, then any line of S has exactly two points. (ii) S is an affine plane, an affine plane with one improper point, or a punctured projective plane. (iii) S is the Fano-quasi -plane.

Real and Complex Singularities

2016

In this paper a Minkowski analogue of the Euclidean medial axis of a closed and smooth plane curve is introduced. Its generic local configurations are studied and the types of shocks that occur on these are also determined.

Theta-characteristics on singular curves

2007

On a smooth curve a theta–characteristic is a line bundle L with square that is the canonical line bundle ω. The equivalent conditionHom(L, ω) ∼= L generalizes well to singular curves, as applications show. More precisely, a theta–characteristic is a torsion–free sheaf F of rank 1 with Hom(F , ω) ∼= F . If the curve has non ADE–singularities then there are infinitely many theta–characteristics. Therefore, theta–characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta–characteristics (i.e. F with h(C,F) ≡ 0 resp. h(C,F) ≡ 1 modulo 2) in terms of the geometric genus of the curve and certain discrete invariants of a …

Logarithmic Vector Fields and the Severi Strata in the Discriminant

2017

The discriminant, D, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the Severi strata. The smallest is the δ-constant stratum, D(δ), where the genus of the fibre is 0. It is well known, by work of Givental’ and Varchenko, to be Lagrangian with respect to the symplectic form Ω obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to Ω, and mor…