Plane curves whose general points are galois points inthissection,weprovetheorem1. Kim let x be a smooth compact riemann surface or a smooth projective curve of genus g. The term supersingular has nothing to do with singular points of curves, and all supersingular elliptic curves are non singular. Singular points of plane curves london mathematical society student texts book title.
Computing singular points of projective plane algebraic. Plane curves whose singular points are cusps hisao yoshihara communicated by louis j. Determining plane curve singularities from its polars. In either case, there is a smooth map from r to the plane which defines the curve in the. Introduction we study semigroups associated to singular points of plane curves. Computing singular points of plane rational curves. We compute the singular points of a plane rational curve. The precise definition of a singular point depends on the type of curve being studied. The problem of determining the maximum number of singular points on rational homology projective planes with quotient singularities is related to the algebraic montgomeryyang problem 19, 11. Convex curves and their characterization, the four vertex theorem. Let a be the sum of the milnor numbers of the singular.
Let c be an irreducible curve of degree d in the complex projective plane. This text introduces readers to an attractive area of mathematics that uses techniques from algebra, algebraic geometry. The narcissists of book singular points of plane curves 2004 and surplus in handling likely srebreniks. We study equisingular strata of plane curves with two singular points of prescribed types. On the enumeration of complex plane curves with two. Computing singular points of plane rational curves request pdf. The method of the previous work is generalized to this case. Pdf on the enumeration of complex plane curves with two. We believe that our paper gives a nontrivialcontribution to the understanding of the existence of certain types of weierstrass points and osculating points on these curves. We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blowup x of the projective plane along their singular points. A wolfram language approach to real numerical algebraic. Journal of symbolic computation 43 2008 92117 in this paper, we improve the method by chionh and sederberg by giving an ef. One question is is it possible to parametrize the curve. We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve.
The method of the previous work 25 is generalized to this case. In i take a constructive point of view and show instead that a curve of degree with or more singular points is reducible. In particular, the algorithm applies to describe the equisingularity class of a germ of plane curve by giving this information combinatorially encoded by means of an enriques diagram from the enriques. Singular plane curves with infinitely many galois points. My goal is understand that the category of nonsingular projective curves is equivalent to the finite generated field extension of k of transendence degree 1. Singularity theory of plane curves and its applications abstract. There exists an integer q such that the intersection multiplicity irc,trc q for all smooth point r. This unit and the following one are devoted to the study of curves in low dimensional spaces. In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The text covers all the essentials in a style that is detailed and expertly written by one of the foremost researchers and teachers. Symbolic computation 1990 9, 405421 singular points of algebraic curves takis sakkalis and rida farouki ibm research division, thomas dr.
For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. Pdf singular points of plane curves semantic scholar. Singular points of plane curves london mathematical. Singular points of plane curves london mathematical society student texts this book has arisen from the authors successful course at liverpool university. First, the problem for two ordinary multiple points of fixed multiplicities is solved.
On the number of singular points of plane curves core. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Box 218, yorktown heights, new york 10598, usa received 28 april 1988 given an irreducible algebraic curve fx,y. The algorithm involves the combined applications of homotopy continuation methods and a method of. The term supersingular has nothing to do with singular points of curves, and all supersingular elliptic curves are nonsingular. Even the simplest singularities of planar curves, e. We now show that algebraic curves can be parametrized locally near nonsingular points. Given an irreducible algebraic curve fx,y0 of degree n. On the enumeration of complex plane curves with two singular. The complex elliptic curves with complex multiplication. For a survey and the history of the subject up to 1986. P2 a galois point with respect to c if the point projection from p induces a.
Weierstrass points and ramification loci on singular plane. Chapter 7 local properties of plane algebraic curves risc. Gap orders of rational functions on plane curves with few. C download it once and read it on your kindle device, pc, phones or tablets. A classicaltopic of study in complex analysis and algebraic geometry was the study of weierstrass points of x. Yet applications of singularity or catastrophe theory to pdes. In the argument, we produce a polynomial of smaller degree that meets the given curve in too many points, so has a common factor with the given.
Numerous and frequentlyupdated resource results are available from this search. Pdf on the number of singular points of plane curves. We assume that c is not smooth and each singular point is a cusp i. A singular point p of multiplicity r on an affine plane curve c is called ordinary iff the r tangents to c at p are distinct, and nonordinary otherwise. Computing singular points of plane rational curves falai chena, wenping wangb,yangliub. When we have a singular curve cin projective space, we will treat cits normalization, i. This book has arisen from the authors successful course at liverpool university. In thefirstsectionwe make easy extensions ofk2, th. On hodge theory of singular plane curves 3 in particular, one has the following formulas for the rst betti number of c. Semigroups associated to singular points of plane curves. We compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the. Letefcpf, f be an irreducible polynomial, so 0 is the equation for an irreducible plane curve. Gap orders of rational functions on plane curves with few singular points.
Weierstrass points and ramification loci on singular plane curves. The newton polygon of plane curves with many rational points. Singular points of plane curves london mathematical society. Now we consider the case of a curve chaving several irreducible components. C t c wall singularities of plane curves are best understood in terms of complex numbers. Our base field k is assumed to be algebraically closed. In particular, we consider the enumerative problem for plane curves with two singular points of linear singularity types. Pdf this is an extended, renovated and updated report on a joint work which the second named author presented at the conference on algebraic geometry. We prove that certain integers n cannot occur as degrees of linear series without base points on the normalization of a plane curve whose only singularities are a small number of nodes and ordinary cusps. An irreducible curve of degree has at most complex projective singular points.
Request pdf singular plane curves with infinitely many galois points for a plane curve c, we call a point p. We assume that each singular point is a one place point with multiplicity 2 or 3. The algorithms require only rational arithmetic operations on the coefficients of fx,y0, and avoid the need. Topological questions about symplectic 4manifolds 2. Research article computing singular points of projective. Preface the study of singular points of algebraic curves in the complex plane is a meeting point for many di.
Geometry of algebraic curves university of chicago. As a conclusion we give examples where and how the theorem works, and that it. Decomposition of the link complement and the milnor fibre 10. We remark that if a rational homology projective plane s is allowed to have. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter. Examples of applications of newton polygon to the theory of singular points of algebraic functions, trans. We also outline the algorithm on computing the singular points of projective plane algebraic curves, and a erwards we analyze feasibility andcomplexityofthealgorithm.
Weierstrass points and ramification loci on singular plane curves by e. Cambridge core discrete mathematics information theory and coding singular points of plane curves by c. Algebraic curves in the plane may be defined as the set of points x, y satisfying an equation of the form f x, y 0, where f is a. A wolfram language approach to real numerical algebraic plane. Computation of singular points in this section, a method of solving the overdetermined polynomial systems is presented rst. The algorithm involves the combined applications of homotopy continuation methods and a method of root. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the milnor numbers of its singular points. Some nonsingular and nodal curves in the plane and other rational surfaces, blowing up the nodes. It comes from the phrase singular values of the jinvariant used for values of the jinvariant for which a complex elliptic curve has complex multiplication. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter values of the singular points. We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters.
The full treatment uses techniques from algebra, algebraic geometry, complex analysis and topology and makes an attractive chapter of mathematics, which can be used as an introduction. Let c be an irreducible curve of degree d in the complex projective plane p2. Charles terence clegg wall, mathematiker grossbritannien. A rational curve pt can be represented implicitly by fx,y,w 0, for a unique.
As a consequence we compute the gonality of such a curve. The study of singular points of algebraic curves in the complex plane is a meeting point for many di. We explain classical hyperbolic differential geometry on plane curves. Use features like bookmarks, note taking and highlighting while reading singular points of plane curves london mathematical society student texts book 63. Cambridge university press 0521839041 singular points of. Baker, examples of applications of newton polygon to the theory of singular points of algebraic functions, trans. Since this will only increase r, we may assume v h0x,l. Pellikaan, on the klein quartic, the fano plane and curves representing designs, in the blahut festschrift. Cp2 is nodal if every singular point of chas a quadratic tangent cone consisting of two distinct lines. Journal of symbolic computation 43, 92117, 2008, which are based on the. Let 0, 0 be a singular point of 0 and well assume thathas two branches at 0, 0, that is, f f i 2 is the irreducible decomposition ofin kx, f. During the nineteenth and early twentieth century algebraic geometers working on plane curves developed methods which allowed them to deal with singular curves. Let 0, 0 be a singular point of 0 and well assume thathas two branches at 0, 0, that is, f f i 2 is the irreducible decomposition of in kx, f.
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