Limits of PGL(3)-translates of plane curves, I

un 2 00 7 LIMITS OF TRANSLATES OF PLANE CURVES , I

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of a subscheme, determined by the curve, of the P 8 of 3 × 3 matrices. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in ...

Duality of Plane Curves

The dual of an algebraic curve C in RP defined by the polynomial equation f(x, y, z) = 0 is the locus of points ( ∂f ∂x (a, b, c) : ∂f ∂y (a, b, c) : ∂f ∂z (a, b, c) ) where (a : b : c) ∈ C. The dual can alternatively be defined geometrically as the image under reciprocation of the envelope of tangent lines to the curve. It is known that the dual of an algebraic curve is also an algebraic curve...

Vertices of Plane Curves

PROOF. Apply Theorem J for the case w==r. We obtain for gr an expression which differs from the one just written only in the fact that the terms Brtr+ifr+i+Brtr+2fr+2+ • • • +Br,nfn are missing from its numerator. But the coefficients J3r,, = 0 when r<s. Hence the two expressions are equal. REMARK. The generalization of the method to orthonormalization with respect to a general norming or weigh...

Linear Systems of Plane Curves

Sparse Parametrization of Plane Curves

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.