Characterizing singular curves in parametrized families of biquadratics
نویسندگان
چکیده
We consider families of biquadratic curvesB = 0 onC, defined with respect to arbitrarily many complex parameters. Due to the fact that these families include curve intersections across different parameter combinations, they represent a generalization of the non-intersecting foliations of one-parameter invariant curves associated with the QRT mapping. We use algebraic methods involving discriminants to provide a complete classification of the singular curves in these families. In developing this classification, we exploit the special symmetric nature of B; namely, that it is a quadratic in x and y whose reflection in the line y = x is given by a simple change of parameters. We also define a range of conditions in the biquadratic’s parameters and demonstrate the manner in which they correspond to different geometric realizations of the singular curves. PACS numbers: 02.30.Ik, 02.40.Xx, 02.40.−k
منابع مشابه
On the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کاملPeriodic Attractors versus Nonuniform Expansion in Singular Limits of Families of Rank One Maps
We analyze parametrized families of multimodal 1D maps that arise as singular limits of parametrized families of rank one maps. For a generic 1-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by...
متن کاملCounting Nodes on Rational Plane Curves
In this paper, we consider polynomial parametrized curves in the affine plane k over an algebraically closed field k. Such curves are given by κ : k → k : t 7→ (x(t), y(t)), and may or may not contain singular points. The problem of how many singular points there are is of specific importance to the theory of polynomial knots, as it gives a bound on the degrees necessary to achieve a parametriz...
متن کاملRemarks on Families of Singular Curves with Hyperelliptic Normalizations
We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus g ≥ 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is “vector bundle-free” and relies on deformation theory and bending-and-breaking of rational curves in Sym(S). We also give examples of fa...
متن کاملThe Geometry of Equisingular and Equianalytic Families of Curves on a Surface
Introduction The existence of singular algebraic curves with given invariants and given set of singular-ities and, at the same time, the study of (equisingular) families of such curves is a very old, but still attractive and widely open, problem. Already, at the beginning of the 20th century, the foundations were made in the works of Pl ucker, Severi, Segre and Zariski. In the sequel the theory...
متن کامل