METU Fall 1999 Math 459 Introduction to Topology of Real Algebraic Curves

MIDTERM I

NOVEMBER 19, 1999

Problem 1. (10 pts)

(1) How many ovals should have an M-curve of degree 8 ?

(2) How deep can be a nest of ovals of a degree 12 curve ?

(3) Why a sextic cannot have more then one nest ?

Problem 2. (30 pts)

(1) Find the equation of a line, ℓ, in ℝP^{2} tangent at the point [−1 : 1 : 1] to a cubic A = {2x_{0}^{3} + x_{0}^{2}x_{1} + x_{1}x_{2}^{2} = 0}.

(2) Find the intersection points of ℓ with each of the 3 coordinate lines in ℝP^{2}.

(3) Determine the coordinate system in ℝP^{2}, in which ℓ is the infinity line and the point [1 : 0 : −1] is the center of coordinates.

(4) Determine the equation of A in the new coordinate system.

(5) Determine the singular points of A.

Problem 3. (30 pts)

(1) Determine the singular points of a curve A = {x^{3}y + y^{3} = 0} ⊂ ℝ^{2}

(2) Determine the equation of the projective closure, Ā, of A in ℝP^{2}.

(3) Determine the points of Ā at infinity.

(4) Determine if the points of Ā at infinity are singular.

(5) If A has singular points at infinity, then write down the local equations defining them (in a chart with the origin at a singularity).

(6) Sketch the locus of A near the singular points and determine their type.

Problem 4. (30 pts) Consider a rational function f(x, y) = \frac{x^{2}+2y^{4}+1}{x^{2}−2y^{2}}.

(1) Define f as a function f: ℝ^{2} → ℝP^{1}. Check, that f is well-defined for any (x, y) ∈ ℝ^{2}.

(2) Will f be also well-defined if we try to extend f to a mapping from ℂ^{2} to ℂP^{1}, using the same formula as in (1) ?

(3) Try to extend f to a function F : ℝP^{2} → ℝP^{1}.

(4) Check if F is well-defined at all the points at infinity.

(5) Give a formula for the restriction ℝP^{1} → ℝP^{1} of F to the infinity line ℓ ≅ ℝP^{1}.

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