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Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.

Francisco Santos (1997)

Revista Matemática de la Universidad Complutense de Madrid

We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...

Perturbing plane cruve singularities.

Eduardo Casas-Alvero, Rosa Peraire (2003)

Revista Matemática Iberoamericana

We describe the singularity of all but finitely-many germs in a pencil generated by two germs of plane curve sharing no tangent.

Pinceaux de courbes planes et invariants polaires

Evelia R. García Barroso, Arkadiusz Płoski (2004)

Annales Polonici Mathematici

We study pencils of plane curves f t = f - t l N , t ∈ ℂ, using the notion of polar invariant of the plane curve f = 0 with respect to a smooth curve l = 0. More precisely we compute the jacobian Newton polygon of the generic fiber f t , t ∈ ℂ. The main result gives the description of pencils which have an irreducible fiber. Furthermore we prove some applications of the local properties of pencils to singularities at infinity of polynomials in two complex variables.

Points rationnels de la courbe modulaire X 0 ( 169 )

Jean-François Mestre (1980)

Annales de l'institut Fourier

On démontre que les seuls points rationnels sur Q de la courbe X 0 ( 169 ) sont les pointes.En conséquence, il n’existe pas de courbe elliptique définie sur Q possédant un sous-groupe cyclique rationnel d’ordre 13 2 .

Puiseux Expansion of a Cuspidal Singularity

Maciej Borodzik (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an effective and elementary method of determining the topological type of a cuspidal plane curve singularity with given local parametrization.

Quasipositivity and new knot invariants.

Lee Rudolph (1989)

Revista Matemática de la Universidad Complutense de Madrid

This is a survey (including new results) of relations ?some emergent, others established? among three notions which the 1980s saw introduced into knot theory: quasipositivity of a link, the enhanced Milnor number of a fibered link, and the new link polynomials. The Seifert form fails to determine these invariants; perhaps there exists an ?enhanced Seifert form? which does.

Rational equivalence on some families of plane curves

Josep M. Miret, Sebastián Xambó Descamps (1994)

Annales de l'institut Fourier

If V d , δ denotes the variety of irreducible plane curves of degree d with exactly δ nodes as singularities, Diaz and Harris (1986) have conjectured that Pic ( V d , δ ) is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that Pic ( V d , 1 ) is a finite group, so that the conjecture holds for δ = 1 . Actually the order of Pic ( V d , 1 ) is 6 ( d - 2 ) d 2 - 3 d + 1 ) , the group being cyclic if d is odd and the product of ℤ 2 and a cyclic group of order 3 ( d - 2 ) ( d 2 - 3 d + 1 ) if d is even.

Semi-group conditions for affine algebraic plane curves with more than one place at infinity.

Penelope G. Wightwick (2007)

Revista Matemática Complutense

An interesting and open question is the classification of affine algebraic plane curves. Abhyankar and Moh (1977) completely described the possible links at infinity for those curves where the link has just one component, a knot. Such curves are said to have one place at infinity. The Abhyankar-Moh result has been of great assistance in classifying those polynomials which define a connected curve with one place at infinity. This paper provides a new proof of the Abhyankar-Moh result which is then...

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