Real plane algebraic curves with prescribed singularities. (English) Zbl 0845.14017

Shustin, Eugenii

Topology 32, No. 4, 845-856 (1993).

There is suggested a construction of plane algebraic curves with prescribed singularities, which applies both in the real and complex case. The construction is based on the Viro method of gluing polynomials and on the geometry of equisingular families of plane curves. As application it is proved that
(1) for any nonnegative integers \(a,b,c\) satisfying \(a + b + 2c = (d-1)(d - 2)/2\) there exists an irreducible real nodal plane curve of degree \(d\) with \(a\) real nodes, \(b\) isolated points and \(c\) pairs of imaginary conjugate nodes;
(2) for any nonnegative integers \(k,d\) satisfying \(k \leq (d^2 - 3d + 4)/4\), there exists an irreducible real plane curve of degree \(d\) with \(k\) real ordinary cusps.

Reviewer: E.I.Shustin (Tel Aviv)

MSC:

14H20	Singularities of curves, local rings
14P05	Real algebraic sets

Keywords:

plane algebraic curves with prescribed singularities