Singular plane curves: freeness and combinatorics

In this wor we introduce the definition of fully simple singularities of parameterized curves and explain that this definition is more natural than the definition of simple singularities. The set of fully simple singularities is much smaller than the set of simple ones. We determine and classify all fully simple singularities of plane and space curves, with any number of components. Our classification results imply that any fully simple singularity of a plane or a space curve is quasi-homogeneous (whereas there is a number of non-quasi-homogeneous simple singularities). Another outcome of our classification results is a one-to-one correspondence between the fully simple singularities of plane curves and the classical A-D-E singularities of functions.

Let X be a smooth compact RIemann surface (or a smooth projective curve) of genus g. A classicaltopic of study in Complex Analysis and Algebraic Geometry was the study of Weierstrass points of X. For a survey and the history of the subject up to 1986, see [EH]. For another survey containing the main definitions and results on Weierstrass points on singular Gorenstein curves, see [G]. For the case of a base field with positive characteristic,see [L]. Since Weierstrass points are special points on a curve, they have been very useful to study moduli problems. In particular, some subvarieties of the moduli space of smooth genus g curves are defined by the existence of suitable Weierstrass points. Several papers were devoted to the study of Weierstrass points on some interesting classes of projective curves (e.g. smooth plane curves and k-gonal curves). Our paper belong to this set of papers. We consider singular plane curves with ordinary cusps or nodes as only singularities.We believe that our paper gives a non-trivial contribution to the understanding of the existence of certain types of Weierstrass points and osculating points on these curves. In the firstsection we make easy extensions of [K2], Th. 1.1, to the case of singular curves. In the second section we use deformation theory to show the existence of several pairs (C,P) with C in integral nodal plane curve, P e Creg, such that the tangent line D of C at P has high order of contact with C at P (see Theorems 2.1 and 2.2 for precise statements). Calling X the normalization of C and seen Pas a point of X, these pairs (C,P) satisfiesthe conditions of Proposition 1.1 below and hence P is a Weierstrass point of X. In the third section we consider the Weierstrass semigroup of the Weierstrass points obtained in this way. Here the main aim is to give a recipe to extract from the numerical calculations in [K2] as much informations as possible for the Weierstrass semigroup of the pair (X,P). The case of a total inflection point for nodal plane curves was considered in detailsin [CK]. Our recipe (see 3.1) gives

This chapter is devoted to the study of plane curve singularities. First of all, from the existence of the normalization, and the fact that one-dimensional normal singularities are smooth, which are results proved in the previous chapter, it follows that one can parameterize curve singularities. Thus, if R is the local ring of an irreducible plane curve singularity, we have an injection R ⊂ ℂ{t}. From the Finiteness of Normalization Theorem 1.5.19 it follows without much difficulty that the codimension of R in ℂ{t} (as ℂ-vector space) is finite. This codimension is called the i-invariant. Here we prove the converse of this statement, that is, given a subring R of ℂ {t} of finite codimension, generated by two elements, say x(t) and y(t), then R the local ring of an irreducible plane curve singularity, say given by f(x,y) = 0. In order to obtain information on the order of f, we introduce the intersection multiplicity of two (irreducible) plane curve singularities (C, o) and (D, o). This number can be calculated as follows. Consider the normalization ℂ{t} of O c,o , and let (D, o) be given by g = 0. Via the inclusion O C,o ∈ℂ{t}, we can consider at g as an element of ℂ{t}. Then the intersection multiplicity is given by the vanishing order of g. From this it easily follows that if R = ℂ{x(t),y(t)}, and f ∈ ℂ{x,y} is irreducible with f(x(t),y(t)) = 0, then the order of f in y is equal to ord t (x(t)), and similar for x. As an application we prove that if we have a convergent power series f ∈ ℂ {x,y}, which is irreducible when considered as a formal power series, then it is already irreducible as a convergent power series.

In general the singular locus of such a surface is one-dimensional, with at most two components. A transverse slice x = C (where C is a small nonzero constant) cuts out a singular plane curve. The Milnor fiber of this curve undergoes a monodromy transformation when C loops around the origin; the action on its homology groups is called the vertical monodromy. In this article we show how to explicitly calculate this monodromy. Our formula is expressed recursively, by associating to our surface two related quasi-ordinary surfaces which we call its truncation S1 and its derived surface S′, and then expressing the vertical monodromy of S via the monodromies of S1 and of S′. As is well known, there is another fibration over a circle, called the Milnor fibration; here the action on homology is called the horizontal monodromy. In the course of working out our recursion for vertical monodromy, we have discovered what appears to be a new viewpoint about the horizontal monodromy, expressed in a similar recursion which again invokes the same two associated surfaces. In fact this recursion makes sense even outside the quasi-ordinary context, and thus we have found a novel way to express the monodromy associated to the Milnor fibration of a singular plane curve. We begin by working out this situation, to motivate our later setup and to provide a model for the more elaborate calculation. As a corollary to our formulas, we have found that from the vertical monodromies (one for each component of the singular locus), together with the surface monodromy formula worked out in [11] and [4], one can recover the complete set of characteristic pairs of a quasi-ordinary surface. Since these data depend only on the embedded topology of the surface, we thus have a new proof of Gau’s theorem [3] in the 2-dimensional case. As another application, we can employ a theorem of Steenbrink [13] (extended to the non-isolated case by M. Saito [12]) which relates the horizontal and vertical monodromies to the spectrum of the surface and to the

We start with a brief account of knot theory for the following two reasons. First, the links of (plane) curve singularities—which are usually regarded as the simplest class of singularities to investigate—form a special class of knots, the so-called algebraic links. Second, many of the fundamental concepts related to the local topology of a higher dimensional IHS (e.g., Seifert matrix, intersection form, Milnor fibration, Alexander polynomial) have been considered first in relation to knot theory.

This book provides a comprehensive and self-contained exposition of the algebro-geometric theory of singularities of plane curves, covering both its classical and its modern aspects. The book gives a unified treatment, with complete proofs, presenting modern results which have only ever appeared in research papers. It updates and correctly proves a number of important classical results for which there was formerly no suitable reference, and includes new, previously unpublished results as well as applications to algebra and algebraic geometry. This book will be useful as a reference text for researchers in the field. It is also suitable as a textbook for postgraduate courses on singularities, or as a supplementary text for courses on algebraic geometry (algebraic curves) or commutative algebra (valuations, complete ideals).

We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over (ℚ) or finite fields.

We study the matrix factorizations defined by the generic plane projections of a curve singularity of $$(\mathbb {C}^n,0)$$ ( C n , 0 ) . On the other hand, given a plane curve singularity $$(Y,0)\subset (\mathbb {C}^2,0)$$ ( Y , 0 ) ⊂ ( C 2 , 0 ) we study the family of matrix factorizations defined by the space curve singularities $$(X,0)\subset (\mathbb {C}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) such that (Y, 0) is the generic plane projection of (X, 0).

Plane curves specified parametrically by y=sp, x=sq, p and q integral, p>or=q>0 are either graphs of monomials, or have a singularity at the origin which can be one of four types: cusp, bend, kink or end. These types are classified by (p,q)=(even, odd) or by a winding number index. Singular sections of the cuspoid bifurcation sets, in which all but two of the control variables are set to zero, are either cusps, bends or one of four more-singular curves derived from the above by the addition of the tangent at the origin. An explicit classification is given of the curve types arising in codimension up to 7. Only in such sections can the equations of the bifurcation sets generally be written explicitly.

Singular plane curves with infinitely many Galois points

Determining plane curve singularities from its polars

We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

We generalize and make rigorous a construction by Enriques which allows one to obtain a plane curve as the projection of a non singular curve spanning ℙ4 we show that every non singular curve in ℙr projecting onto a given plane curve can be obtained by the same construction. Finally we prove that every non singular plane curve of degree d is the projection of a (non singular) curve of degree 2d-1 spanning ℙ4, and that no lower degree is possible.

On Singularities of Plane Curves Given by Parametric Equations

1303. On C. N. Srinivasiengar's Note on Singularities of Plane Curves Given by Parametric Equations