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

PROBLEM LIST II

DECEMBER 6, 1999

Real and Complex Schemes of Real Algebraic Curves

Problem 1. Sketch and give the codes of the real schemes for the three rigid isotopy classes of M-curves of degree 6. Sketch their complex schemes.

Problem 2. Sketch the real schemes and give the codes for several (at least 10) L-curves of degrees 5 and 6. For L-curves of type I describe the complex scheme.

Problem 3. Construct M-curves of degrees 7 and 8 by methods of Harnack and Hilbert. Code their real schemes and describe their complex schemes.

The Euler Characteristic

Problem 4. Determine the Euler characteristic of (1) the letters of the Alphabet,

(2) a surface F_{g} with p holes,

(3) a solid body bounded in ℝ^{3} by a surface F_{g},

(4) n-dimensional ball B^{n} and sphere S^{n} (n = 3, 4, …).

Problem 5. Two surfaces, S_{1} and S_{2} are obtained from a genus 3 orientable surface F_{3} after cutting along a pair of disjoint circles. What can be the surfaces S_{1} and S_{2} ?

Problem 6. Which surface we may obtain if we cut N_{k} along (1) one-sided curve ? (2) two-sided curve ? Consider the cases k = 2, 3.

Problem 7. p: X → Y is an n-fold branching covering of surfaces X and Y, so, p^{−1}(y) contains n points for the usual points y ∈ Y and less then n for the branching points y. Consider the case of double branching points, so that p^{−1}(y) contain not less then (n − 1) points for any y.

Explain the Hurwitz formula

χ(X) = nχ(Y) − k

where k is the number of branching points.

How this formula changes if we allow branching points of higher multiplicity ?

Problem 8. Determine the number of tangent lines from a generic point on a plane to the (complex) curve of degree d.

Hint: Consider the central projection from a complex projective curve, ℂA, to a complex line, ℂP^{1} and apply the result of the previous problem.

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