Simple singularities of parametrized plane curves in positive characteristic

Let K be an algebraically closed field of characteristic p>0. The aim of the article is to give a classification of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of singularities which are not simple such that almost all singularities deform to one of those and show that remaining singularities are simple.     

Recursions for characteristic numbers of genus one plane curves

Characteristic numbers of families of maps of nodal curves toP ² are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.     

Geometry of plane curves

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.     

Real plane algebraic curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves.     

Cubic curves in the triangle plane

The subject of this paper are two pencils of cubic curves that are the result of certain geometrical constructions in the triangle plane. One of them turns out to be the probably most significant pencil of anallagmatic cubics that are associated with triangle geometry. Both contain virtually all important single cubics, and other well known curves appear closely connected with them.Dedicated to Professor T.G. Ostrom on the occasion of his 80th birthday     

Intermediate links of plane curves

For a smooth complex curve C ? ?² we consider the link L_r = C ∩ ?B_r, where B_r denotes an Euclidean ball of radius r > 0. We prove that the diagram D_r obtained from L_r by a complex stereographic projection satisfies χ(C ∩B_r) = rot(D_r)?wr(D_r). As a consequence we show that if D_r has no negative Seifert circles and L_r is strongly quasipositive and fibered, then the Yamada–Vogel algorithm applied to D_r yields a quasipositive braid.     

On multi-Frobenius non-classical plane curves

In this paper, we characterize the plane curves over \mathbb F_q{\mathbb {F}_q} which are Frobenius non-classical for different powers of q.     

On maximal curves in characteristic two

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}} ^t y ^{q /2 i} =x ^{q +1}, q=2 ^t , provided that q/2 is a Weierstrass non-gap at some point of the curve. Received: 3 December 1998     

Singularities of plane algebraic curves

We give an exposition of some of the basic results on singularities of plane algebraic curves, in terms of polynomials and formal power series.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Intersection graphs of curves in the plane

Let V be a set of curves in the plane. The corresponding intersection graph has V as the set of vertices, and two vertices are connected by an edge if and only if the two corresponding curves intersect in the plane.It is shown that the set of intersection graphs of curves in the plane is a proper subset of the set of all undirected graphs. Furthermore, the set of intersection graphs of straight line-segments is a proper subset of the set of the intersection graphs of curves in the plane. Finally, it is shown that for every k ≥ 3, the problem of determining whether an intersection graph of straight line-segments is k-colorable is NP-complete.