Summary: Sample solutions for Problems 7, 8 and 10 in Problem Set 12 with
Chris Peterson and Hirotachi Abo
Problem 7, 8 and 10 (Set 12). Let C be an irreducible plane curve in P2
over an algebraically closed field k, that is, the zero locus of an irreducible
polynomial F k[x, y, z]. Recall that the tangent line Tp(C) to C at a point
p C is defined by the following single polynomial equation:
xFx(p) + yFy(p) + zFz(p) = 0.
This allows us to define the map from C to the dual projective space (P2
(p) = the dual of Tp(C) = [Fx(p) : Fy(p) : Fz(p)].
be the projective closure of (C) in (P2
. This C
is called the dual
curve of C.
Question. How can we compute the homogeneous polynomial whose zero
locus is the dual curve C