Find the singular points and tangent lines at the singular

Question

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.