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

MIDTERM II

DECEMBER 18, 1999

Problem 1. (25 pts) (a) Sketch a singularity defined near the origin by each of the following equations: x^{13} + y^{8} = 0, x^{12} + y^{7} = 0, x^{11} + y^{7} = 0.

(b) Give the equations and sketch the simple singularities of types A_{n}, D_{n}, E_{6}, E_{7}, E_{8}.

Problem 2. (25 pts) (a) An orientable closed surface F of genus 4 is cut along a simple closed curve, γ, so that F ∖ γ splits into two connected components. What can be these components ? Sketch a curve γ on a surface F for every possibility you have found.

(b) A surface F from (a) is cut along a pair of disjoint simple closed curves, α and β, so that the complement F ∖ (α ∪ β) is connected. What is this complement ?

In the answers to (a) and (b) the surfaces should be characterized as “genus n surfaces, F_{n}, with k holes”, or “non-orientable surfaces N_{n} with k holes”, so, your task is to determine n and k in each case.

Problem 3. (20 pts) (a) Prove that the surface F_{g} (closed, orientable of genus g) cannot be a double branched covering over a homeomorphic surface, F′_{g}, for g ⩾ 2.

(b) What can you say about the case g = 0 and g = 1 ? If such a double branched covering exists, what is the number of the branch points ?

Problem 4. (35 pts) (a) Describe the real and complex schemes non-singular real plane algebraic curves of type I, which have degrees ⩽ 5.

(b) Give a construction of Harnack’s and Hilbert’s M-curves of degree 6. Show the complex orientations of these curves.

(c) Sketch three examples of L-curves of degree 6 of type I, with different real schemes. Show the complex orientations and give the codes of these schemes.

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