Reference no: EM131084020
Math 176: Algebraic Geometry, Fall 2014- Assignment 6
1. Let F be an affine plane curve with deg(F) = n. Show that if P ∈ V(F) has multiplicity n, then F consists of n lines through P (not necessarily distinct).
2. Find the singular points and tangent lines at the singular points for each of the following affine plane curves:
(a) x3 + y3 - 3x2 - 3y2 + 3xy + 1
(b) y2 + (x2 - 5)(4x4 - 20x2 + 25)
3. Suppose X,Y ⊆ An be varieties and Y 
X. Prove dim(Y) < dim(X).
4. A simple point P on a curve C = V(F) is called a flex if ord(L) in OP(V(F)) is at least 3, where L is the tangent line to F at P. The flex is called ordinary if ord(L) = 3, and is a higher flex otherwise.
(a) Let F(x, y) = y - xn ∈ k[x, y]. For which n does F have a flex at P = (0, 0), and what kind of flex?
(b) Suppose P = (0, 0), L = y is a tangent line, F = y + ax2 + · · · . Show that P is a flex if and only if a = 0. Give a simple criterion for calculating ord(y) and therefore for determining if P is a higher flex.
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