Math 662: Elliptic Curves
HW #2
February 15, 2010
Due Tuesday, February 23.
1. Let C : F (X, Y, Z) = 0 be a curve in P2 (over an algebraically closed field K). Show
that the following polynomial identity holds:
∂F
∂F
∂F
X+
Y +
Z = (deg F ) · F.
∂X
∂Y
∂Z
Let P ∈ C be a smooth point. Show that the tangent line to C at P is given by
∂F
∂F
∂F
(P )X +
(P )Y +
(P )Z = 0.
∂X
∂Y
∂Z
Be sure to check that the quantities in this equation are well-defined independent of
the choice of coordinates of P .
2. Let C : F (X, Y, Z) = 0 be a curve in P2 (over an algebraically closed field K). Let L
be a line in P2 , L 6⊆ C. Show that
X
i(P ; L, C) = deg C.
P ∈L∩C
3. Let C : (X 2 + Y 2 )2 − (X 2 − Y 2 )Z 2 be the lemniscate in P2 (C) (r2 = cos(2θ) in polar
coordinates). Compute the multiplicities at the intersection points between C and the
lines X = 0 and Z = 0.