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

Problem List III

Problem 1. A curve of the type I of degree 2k + 1 has α ovals.

(1) Which estimate on α follows from the Rokhlin-Mishachev complex orientation formula ?

(2) What can be said about α if such a curve has real scheme ⟨J⊥⊥α⟩ ?

Problem 2. Prove equivalence of the “index integration” form of the Rokhlin complex orientation formula to its classical formulation.

Problem 3. Consider curves of degree 7 with the real scheme ⟨J⊥⊥1⟨α⟩⊥⊥β⟩. (1) Which values of α and β can you allow ?

(2) What values of α and β can be allowed for the type I curves ? Determine the number of such curves.

(3) Determine the complex schemes of such type I curves using the Rokhlin-Mishachev formula and give their list.

(4) Are there any degree 7 curves which have real schemes other then ⟨J⊥⊥1⟨α⟩⊥⊥β⟩ ? What are they ?

Problem 4. Consider curves of degree 8 with the real scheme ⟨1⟨α⟩⊥⊥β⟩.

(1) What can be the values of α and β allowed by the Harnack, Petrovskii and Arnold inequalities ? What can be said in addition using Rokhlin’s and Gudkov-Krakhnov-Kharlamov’s congruences ?

(2) What can be the values of α and β for M-curves with the above schemes allowed by the Rokhlin congruence ?

(3) What can be the values of α and β for type I curves with the above schemes allowed by the Arnold congruence ?

(4) Which estimate on α, β follows from the Rokhlin complex orientation formula ?

(5) Determine the complex orientations for the M-curves and (M − 2)-curves of type I with the above real schemes using the Rokhlin-Mishachev formula.

Answer similar questions for the curves of degree 8 with a real scheme ⟨1⟨α⟩⊥⊥1⟨β⟩⊥⊥γ⟩.

Problem 5. Deduce the Harnack inequality from the Bezout theorem.

Problem 6. Give several examples of curves of degree 7 with a real scheme ⟨J⊥⊥1⟨3⟩⊥⊥3⟩ which can be distinguished using the Bezout theorem. How many rigidly nonisotopic examples of such curves you can imagine ?

Problem 7. Which real algebraic curves of degree 7 can be obtained after perturbation of the following singular curves using the Viro method ? Give particular constructions of curve with the schemes ⟨J⊥⊥1⟨α⟩⊥⊥β⟩ for

(1) α = 12, β = 2,

(2) α = 1, β = 13.

Problem 8. Consider a singular curve, ℝA, splitting into 4 oriented lines in a generic position in ℝP^{2}. Determine ind_{ℝA}(x) for every point x ∈ ℝP^{2} and evaluate ∫_{ℝP^{2}} ind_{ℝA}(x)^{2} dχ(x).