zbMATH — the first resource for mathematics

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

MSC:
 14H20 Singularities of curves, local rings 14P05 Real algebraic sets
Full Text: