On the Classification of Rational Plane Curves of Type (d,m)

Let C P²=P²(C) be a rational plane curve of degree d and let denote the maximal multiplicity of the singular points of C. We say that C is of type (d, ). Let P C be a singular point, and let r{P} be the number of the branches of C at P. Set (C)= {P Sing(C)}(r_{P}-1). We say that C is of type (d, , ) if C is of type (d, ) and = (C). We classify all rational plane curves of type (d,d-2). We give the complete list of all rational plane curves of type (d,d-2). In particular, we provide an inductive algorithm to construct such curves. Furthermore, we show that any such curve C is transformable into a line by a Cremona transformation. We also construct some classes of rational plane curves of type (d,d-3,1).

Autorentext

A lecturer in Algebraic Geometry, Mathematics department, Faculty of science, Sohag University, Egypt.

Klappentext

Let C P²=P²(C) be a rational plane curve of degree d and let denote the maximal multiplicity of the singular points of C. We say that C is of type (d, ). Let P C be a singular point, and let r{P} be the number of the branches of C at P. Set (C)= {P Sing(C)}(r_{P}-1). We say that C is of type (d, , ) if C is of type (d, ) and = (C). We classify all rational plane curves of type (d,d-2). We give the complete list of all rational plane curves of type (d,d-2). In particular, we provide an inductive algorithm to construct such curves. Furthermore, we show that any such curve C is transformable into a line by a Cremona transformation. We also construct some classes of rational plane curves of type (d,d-3,1).