Middle East Technical University Fall 1999 Math 459 Introduction to Topology of Real Algebraic Curves

PROBLEM LIST I

Homeomorphisms, topological properties, homotopies and isotopies

Problem 1. Classify the letter A, B, ..., up to homeomorphism and up to homotopy equivalence. Which letters are contractible ?

Problem 2. Sketch topologically distinct configurations of (1) a pair of ovals in ℝP^{2},

(2) a pair of lines and an oval in ℝ^{2},

(3) a pair of lines and an oval in ℝP^{2}. In which of these cases the real curves you obtain as the union are

(1) non-homeomorphic ?

(2) homeomorphic, but not ambient homeomorphic ?

(3) homeomorphic, but not homotopic in ℝP^{2} ?

(4) non-isotopic ?

Problem 3. How many connected components are contained in the complements of the configurations in Problem 2 ? Which of these components are not simplyconnected ?

Bezout and Harnack Theorems

Problem 4.

(1) How many ovals should have an M-curve of degree 10 ?

(2) How deep can be a nest of ovals of a degree 10 curve ?

(3) Explain why a curve with a maximal nest cannot be an M-curve (prove that this statement holds for curves of any degree d ⩾ 4).

(4) Explain why a curve of degree 10 (or less) cannot have two nests of ovals, with each nest having depth 3 (or more) ?

(5) Why it cannot have ⩾ 5 nests, with each nest having depth 2 (or more) ?

Projective planes ℝP^{2} and ℂP^{2}, coordinate change

Problem 5.

(1) Find the equation of the line, ℓ, in ℝP^{2} tangent at the point [1 : 1 : 1] to the conic A = {x_{0}^{2} + x_{1}^{2} − 2x_{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 a matrix A ∈ P GL(3, ℝ), which defines a projective transform in ℝP^{2} mapping the original coordinate lines {x_{i} = 0}, i = 0, 1, 2, into the new coordinate lines ?

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

Typeset by AMS-TEX

Problem 6.

(1) Prove that the curve A = {(x^{4} − y^{2})x = 1} ⊂ ℝ^{2} is non-singular.

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

(3) Determine the points of Ā at infinity.

(4) Determine if Ā has singular points at infinity.

Problem 7. Consider a rational function, f, on ℂ, f(z) = \frac{z^{3}+2z−3}{z+4}.

(1) Define f as a function f: ℂ → ℂP^{1} (explain why f is well-defined for any z ∈ ℂ).

(2) Extend f to a function F : ℂP^{1} → ℂP^{1}.

(3) Give a formula for F around the infinity point; i.e., determine function, f_{1}(w), f_{1} : ℂ → ℂ describing F in the other chart of ℂP^{1}.

(4) Find the first two non-trivial terms in the Tailor expansion of F at infinity (i.e., an expansion of f_{1}).

Problem 8. Consider a rational function f(x, y) = \frac{x^{4}+y^{3}}{x^{2}−y+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) Find a point, P, on the infinity line of ℝP^{2} at which F is not well-defined.

(5) Give a formula for F around P; i.e., determine function f_{1} : ℝ^{2} → ℝ describing F in the corresponding charts of ℝP^{2} and ℝP^{1}.

Singularities

Problem 9. Consider mapping f: ℂ → ℂ^{2}, f(z) = (z^{2}, z^{5}).

(1) Determine if the image of f is a non-singular curve A ⊂ ℂ^{2}.

(2) Determine the projective closure, Ā, of A.

(3) Determine if Ā has a singularity at infinity ?

(4) Determine the type of singularities of Ā.

Problem 10. Prove that the image of ℝP^{2} under the Veroneze map is non-singular.