Germs Of Curves Research Articles

The Algebraic Fundamental Group of a Curve Singularity

A classical theorem of Brauner [3], (also Kahler [10], Zariski [15]), gives a formula for the generators and relations of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular curve in C. These elegant formulas depend only on the characteristic pairs of a Puiseux series expansion of the curve. Zariski states this Theorem as part of his discussion on Puiseux series in the chapter on resolution of singularities in his book ”Algebraic Surfaces” [16]. Let γ be the germ of an analyically irreducible plane curve singularity at the origin. Let De be an epsilon ball centered at the origin in C , and let Se be the boundary of De. For e sufficiently small, the pair (De, γ ∩De) is homeomorphic to the pair consisting of the cone over Se and the cone over γ ∩ Se (c.f. Theorem 2.10 [11]). Thus Se − γ ∩ Se is a strong deformation retract of Be − γ ∩ Be, and the topological fundamental group of the knot is isomorphic to π 1 (Be − γ ∩Be). The arithmetic analogue of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular plane curve is thus the algebraic fundamental group π1(Spec(R) − V (f)) , where R = k[[x, y]] is a power series ring over an algebraically closed field k, and f ∈ R is irreducible. The basic theory of the algebraic fundamental group is classical, being understood for Riemann surfaces in the 19th century. The algebraic fundamental group is constructed from the finite topological covers, which are algebraic. Abhyankar extended the algebraic fundamental group to arbitrary characteristic [2] and Grothendieck [8] defined the fundamental group in general. In positive characteristic, Puiseux series expansions do not always exist (c.f section 2.1 of [5]). However, in characteristic zero, the characteristic pairs of the Puiseux series are determined by the resolution graph of a resolution of singularites of the curve germ. As such, the characteristic pairs can be defined in any characteristic from the resolution graph (c.f [5]). In this paper we prove an arithmetic analogue of Brauner’s theorem, valid in arbitrary characteristic. The generators and relations in our Theorem (Theorem 0.1

Metric theory of semialgebraic curves

We present a complete bi-Lipschitz classification of germs of semialgebraic curves (semialgebraic sets of the dimension one). For this purpose we introduce the so-called Holder Semicomplex, a bi-Lipschitz invariant. Holder Semicomplex is the collection of all first exponents of Newton-Puiseux expansions, for all pairs of branches of a curve. We prove that two germs of curves are bi-Lipschitz equivalent if and only if the corresponding Holder Semicomplexes are isomorphic. We also prove that any Holder Semicomplex can be realized as a germ of some plane semialgebraic curve. Finally, we compare these Holder Semicomplexes with Holder Complexes-complete bi-Lipschitz invariant of two-dimensional semialgebraic sets.

Dévissage de la forme de Seifert d'un germe de courbe plane

We propose to unravel the Seifert form of a plane curve germ: from a such Seifert form, we explain how to find the topological type(s) of germs associated with this Seifert form.

THE TOPOLOGICAL ZETA FUNCTION ASSOCIATED TO A FUNCTION ON A NORMAL SURFACE GERM

We associate to a regular function f on a normal surface germ (S, 0) an invariant, called the topological zeta function, which generalizes the same invariant for a plane curve germ; by definition it is a rational function in one variable.We study its poles and their relation with the local monodromy of f, in particular, we prove the “generalized holomorphy conjecture”. We give a formula for this topological zeta function in terms of the log canonical modelof (S, f -1,{0}) , and we also introduce a still more general invariant.

Sur le théorème local de Harnack

Let f be an irreducible germ of a real algebraic plane curve and μ its Milnor number. Let α and c be integers such that α + c ≤ μ 2 . We give a simple construction of a deformation of f with α ovals and c isolated (single) points.

Curves and proximity on rational surface singularities

We study the germs of curves in a rational surface singularity ( S, P) from the point of view of proximity, classifying them up to a notion of equisingularity. We introduce the concept of cluster of infinitely near points and we use it to generalize the Hoskin-Deligne formula, and to give an algorithm to describe a minimal system of generators of a complete ideal in the local ring O s, p.

Species variation in the testicular angiotensin converting enzyme promoter studied in transgenic mice.

We have studied the control of transcription of the testicular angiotensin converting enzyme (ACEt) in normal and transgenic mice. Northern analyses, including a developmental curve and separated germ cells, for ACEt mRNA suggest predominantly post-meiotic expression. Mice transgenic for a construct containing the proximal 298 bp of the rabbit ACEt promoter, with chloramphenicol acetyl transferase (CAT) as a recorder, showed correct tissue regulation while a 86 bp fragment of the promoter led to no expression. Many candidate transacting factor binding elements, previously identified as candidate regulators of transcription driving spermatogenesis, are scattered across this 298 bp in the rabbit (but not the mouse) promoter and may lead to tissue specificity. The recent finding that the proximal 91 bp of the mouse ACEt promoter leads to tissue specific expression of a recorder gene (Howard et al., 1993) emphasizes the difference between the two species and the importance of a cAMP response element (CRE) within this fragment for tissue specific expression. This CRE is conserved in the rabbit promoters we used.

Algebraization of analytic curve germs passing through isolated surface singularities

Algebraization of analytic curve germs passing through isolated surface singularities

A GAGA type theorem on germs of analytic curves through germs of quotient surface singularities

A GAGA type theorem on germs of analytic curves through germs of quotient surface singularities

Simple singularities of space curves

The genesis of this paper lies in theoretical questions in kinematics where a central role is played by naturally occurring families of rigid motions of 3-space. The resulting trajectories are parametrized families of space curves, and it is important to understand the generic singularities they can exhibit. For practical purposes one seeks to classify germs of space curves of fairly small Ae-codimension. It is however little harder to list the A-simple germs, which includes all germs of Ae-codimension ≤11: that then is the principal objective of this paper.

Some remarks on pythagorean real curve germs

In this small note we prove to be true two conjectures proposed in [6], Actually, we obtain a finer theorem, which includes those conjectures and gives, we hope, some additional insight into the study of Pythagoras numbers, even of higher order, of real curve germs.

Note on the degree of $C^0$-sufficiency of plane curves

Let f be a germ of plane curve, we define the 8-degree of sufficiency offto be the smallest integer r such that for any germ g such that j(r)f = j(r)g then there is a set of disjoint annuli in S3 whose boundaries consist of a component of the link of f and a component of the link of g. We establish a formula for the 8-degree of sufficiency in terms of link invariants of plane curves singularities and, as a consequence of this formula, we obtain that the 8-degree of sufficiency is equal to the C° -degree of sufficiency

On the number of branches of a plane curve germ

On the number of branches of a plane curve germ

Pythagorean real curve germs

Let k be a real closed field. A real curve germ over k is a real one-dimensional Noetherian local integral domain with residual field k. A Noetherian local ring A with maximal ideal m and completion  is an AP-ring if for every system of polynomials F∈A[Y]s, Y=(Y1,⋯,Yr), for every formal solution ŷ∈Âr of F=0, and for every integer λ≥0, there exists a solution y∈Ar of F=0 such that y≡ŷ mod mλ Â. A real AP-curve is a real curve germ which is an AP-ring. The Pythagoras number p(A) of A is the least p, 1≤p≤+∞, such that each sum of squares in A is a sum of p squares. The author proves that for any real AP-curve A (over a real closed field) the derived normal ring Ā of A and the completion  of A are real curve germs and p(A)≤p(Â)<∞, p(Ā)=1. The value semigroup of a real AP-curve is a numerical semigroup, that is, an additive subsemigroup of the nonnegative integers, whose complement is finite. The main theorem classifies real AP-curves A which are Pythagorean (that is, p(A)=1) by their value semigroup Γ: Every real AP-curve with value semigroup Γ is non-Pythagorean if and only if there are q,p1,p2 ∈ Γ with q<p1≤p2 such that p1+p2−q∉Γ. Moreover, for a given numerical semigroup Γ the author proves: Every real AP-curve with value semigroup Γ is Pythagorean if and only if for each q∈Γ, p∈E with q<p, one has (q+c)/2≤p. Here c denotes the least positive integer such that Γ contains each p≥c and E is some specified subset of Γ. The paper ends with some applications: A Gorenstein real AP-curve is Pythagorean if and only if its multiplicity is ≤2. A monomial real AP-curve is Pythagorean if and only if it is Arf. There is a list of all Pythagorean real algebroid curves of multiplicity ≤5.