Let c be an irreducible curve of degree d in the complex projective plane p2. The term supersingular has nothing to do with singular points of curves, and all supersingular elliptic curves are nonsingular. Letefcpf, f be an irreducible polynomial, so 0 is the equation for an irreducible plane curve. In either case, there is a smooth map from r to the plane which defines the curve in the. 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. 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. Pellikaan, on the klein quartic, the fano plane and curves representing designs, in the blahut festschrift. Even the simplest singularities of planar curves, e. Singularity theory of plane curves and its applications abstract. We compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the.
The text covers all the essentials in a style that is detailed and expertly written by one of the foremost researchers and teachers. We study equisingular strata of plane curves with two singular points of prescribed types. Cambridge core discrete mathematics information theory and coding singular points of plane curves by c. The algorithm involves the combined applications of homotopy continuation methods and a method of. Computing singular points of plane rational curves falai chena, wenping wangb,yangliub. Yet applications of singularity or catastrophe theory to pdes. On the number of singular points of plane curves core. Preface the study of singular points of algebraic curves in the complex plane is a meeting point for many di.
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. Examples of applications of newton polygon to the theory of singular points of algebraic functions, trans. Journal of symbolic computation 43, 92117, 2008, which are based on the. Chapter 7 local properties of plane algebraic curves risc. This unit and the following one are devoted to the study of curves in low dimensional spaces. Singular points of plane curves london mathematical society student texts this book has arisen from the authors successful course at liverpool university. 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. There exists an integer q such that the intersection multiplicity irc,trc q for all smooth point r.
As a conclusion we give examples where and how the theorem works, and that it. Singular points of plane curves london mathematical society student texts book title. On the enumeration of complex plane curves with two singular. Computing singular points of plane rational curves request pdf. 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. Our base field k is assumed to be algebraically closed. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The method of the previous work is generalized to this case. We compute the singular points of a plane rational curve. Convex curves and their characterization, the four vertex theorem.
Some nonsingular and nodal curves in the plane and other rational surfaces, blowing up the nodes. In thefirstsectionwe make easy extensions ofk2, th. C download it once and read it on your kindle device, pc, phones or tablets. Cambridge university press 0521839041 singular points of. First, the problem for two ordinary multiple points of fixed multiplicities is solved. We assume that c is not smooth and each singular point is a cusp i. We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve. A wolfram language approach to real numerical algebraic plane. Determining plane curve singularities from its polars. Singular points of plane curves london mathematical society. During the nineteenth and early twentieth century algebraic geometers working on plane curves developed methods which allowed them to deal with singular curves. On hodge theory of singular plane curves 3 in particular, one has the following formulas for the rst betti number of c. Singular plane curves with infinitely many galois points.
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. The method of the previous work 25 is generalized to this case. Singular points of plane curves london mathematical society student texts. The precise definition of a singular point depends on the type of curve being studied. Computing singular points of plane rational curves.
Cp2 is nodal if every singular point of chas a quadratic tangent cone consisting of two distinct lines. 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. The algorithm involves the combined applications of homotopy continuation methods and a method of root. We explain classical hyperbolic differential geometry on plane curves. On the enumeration of complex plane curves with two. Journal of symbolic computation 43 2008 92117 in this paper, we improve the method by chionh and sederberg by giving an ef.
Use features like bookmarks, note taking and highlighting while reading singular points of plane curves london mathematical society student texts book 63. Since this will only increase r, we may assume v h0x,l. When we have a singular curve cin projective space, we will treat cits normalization, i. In particular, we consider the enumerative problem for plane curves with two singular points of linear singularity types. Hasegawa journal of algebra 323 2010 10 11 yoshihara 17 proved that rational curves have at most three outer galois points if d 12,24 60. The algorithms require only rational arithmetic operations on the coefficients of fx,y0, and avoid the need. Charles terence clegg wall, mathematiker grossbritannien. An irreducible curve of degree has at most complex projective singular points. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
A rational curve pt can be represented implicitly by fx,y,w 0, for a unique. Given an irreducible algebraic curve fx,y0 of degree n. 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. Pdf on the enumeration of complex plane curves with two. This text introduces readers to an attractive area of mathematics that uses techniques from algebra, algebraic geometry. 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 classicaltopic of study in complex analysis and algebraic geometry was the study of weierstrass points of x. Semigroups associated to singular points of plane curves. As a consequence we compute the gonality of such a curve. Decomposition of the link complement and the milnor fibre 10. 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. It is a corollary of the logarithmic bogomolovmiyaokayaus type inequality due to miyaoka. In i take a constructive point of view and show instead that a curve of degree with or more singular points is reducible.
Weierstrass points and ramification loci on singular plane. Computation of singular points in this section, a method of solving the overdetermined polynomial systems is presented rst. Pdf singular points of plane curves semantic scholar. This book has arisen from the authors successful course at liverpool university. The complex elliptic curves with complex multiplication. Baker, examples of applications of newton polygon to the theory of singular points of algebraic functions, trans. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
Now we consider the case of a curve chaving several irreducible components. Let c be an irreducible curve of degree d in the complex projective plane. Gap orders of rational functions on plane curves with few. For a survey and the history of the subject up to 1986. Let a be the sum of the milnor numbers of the singular. Box 218, yorktown heights, new york 10598, usa received 28 april 1988 given an irreducible algebraic curve fx,y. Plane curves whose singular points are cusps hisao yoshihara communicated by louis j. The newton polygon of plane curves with many rational points. Geometry of algebraic curves university of chicago. C t c wall singularities of plane curves are best understood in terms of complex numbers. Pdf on the number of singular points of plane curves. 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. A wolfram language approach to real numerical algebraic.
Topological questions about symplectic 4manifolds 2. Computing singular points of projective plane algebraic. The term supersingular has nothing to do with singular points of curves, and all supersingular elliptic curves are non singular. Introduction we study semigroups associated to singular points of plane curves. Singular points of plane curves london mathematical. Kim let x be a smooth compact riemann surface or a smooth projective curve of genus g. Symbolic computation 1990 9, 405421 singular points of algebraic curves takis sakkalis and rida farouki ibm research division, thomas dr.
We remark that if a rational homology projective plane s is allowed to have. Plane curves whose general points are galois points inthissection,weprovetheorem1. 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. Research article computing singular points of projective. 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. We also outline the algorithm on computing the singular points of projective plane algebraic curves, and a erwards we analyze feasibility andcomplexityofthealgorithm. The narcissists of book singular points of plane curves 2004 and surplus in handling likely srebreniks. We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters. Numerous and frequentlyupdated resource results are available from this search. We now show that algebraic curves can be parametrized locally near nonsingular points. We assume that each singular point is a one place point with multiplicity 2 or 3. 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. Request pdf singular plane curves with infinitely many galois points for a plane curve c, we call a point p.
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. In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. Weierstrass points and ramification loci on singular plane curves. Weierstrass points and ramification loci on singular plane curves by e. Gap orders of rational functions on plane curves with few singular points. 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. P2 a galois point with respect to c if the point projection from p induces a. 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. One question is is it possible to parametrize the curve.
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