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

FINAL, PART I: TAKE-HOME EXAM (MAKE-UP)

JANUARY 17, 2000

Definition. Given a real algebraic curve ℂA_{0} ⊂ ℂP^{2} with a singular point x ∈ ℂA_{0} we define the genus, g_{x}, of a smoothing of ℂA_{0} at x as follows. Consider a compact 4-ball, B ⊂ ℂP^{2}, around x. If B is sufficiently small, then B ∩ ℂA_{0} is a wedge of r copies of a 2-disc, where r is the number of branches of ℂA_{0} at x. Assume that ℂA_{t}, t > 0, is a small non-singular perturbation of the curve ℂA_{0}. Then B ∩ ℂA_{t} is an oriented connected compact surface with r boundary components, if t is sufficiently small. Its topological type does not depend on the choice of a perturbation and we let g_{x} denote the genus of B ∩ ℂA_{t}.

Problem 1. (20 pts) In the above setting, assuming that t is sufficiently small

(1) determine χ(ℝA_{t} ∩ B) and χ(ℂA_{t} ∩ B). Assume in addition that ℂA_{t} is a perturbation of the type I, i.e., the complement (ℂA_{t} ∩ B) ∖ (ℝA_{t} ∩ B) splits in two connected components, F_{1} and F_{2}.

(2) Note that the closures F̄_{1}, F̄_{2}, of these components are compact surfaces with boundary. What are the boundary components of F_{i} ? How to find their number ?

(3) Determine the Euler characteristic of F̄_{1}, F̄_{2}.

(4) estimate the number, ℓ, of ovals which emerge near x after a perturbation (i.e., find a local version of Harnack’s inequality).

(5) Assuming for simplicity that the singularity is unibranch (i.e., r = 1) find a congruence analogous to Klein’s, i.e., determine the pairity of ℓ.

Information. It is known that the above curve ℂA_{0} can be deformed into a curve ℂB_{0}, so that B ∩ ℂB_{0} is again a union of r copies of 2-discs, but these disc intersect (and self-intersect) transversally, i.e., ℂB_{0} has several nodal singularities emerging near x (see examples on Figure 1 on the Final, part II problem sheet). Denote the number of these nodes by δ_{x}.

Problem 2. (20 pts)

(1) Determine δ_{x} in the examples on Figure 1.

(2) Determine the genus g_{x} in all these examples. (Hint: each nodal point gives a handle aftwer a perturbation. Do not forget also that there are r discs before perturbation).

(3) Using the results of the previous problem, estimate the number ℓ of ovals which can emerge after a perturbation of each of these singularities

Problem 3. (25 pts) What is the maximal number of ovals that may appear after a non-singular perturbation of 7 lines in ℝP^{2}, five of which have a common point ?

(1) Give a list of the real schemes with the maximal number of ovals and present one method of construction in each of the cases.

(2) Does Klein’s congruence allow these curves to be of the type I ?

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Problem 4. (35 pts) A real algebraic curve ℂA of degree 7 has two singular points and no other singularities. Each of these points formed by three non-singular branches tangent to each other (with a simple, quadratic tangency).

(1) Determine χ(ℝA).

(2) Determine χ(ℂA). (Hint: perturb ℂA and use the results of Problems 1 and 2 to determine how the Euler characteristic of ℂA is changing.)

(3) Assuming that the above curve ℂA is of the type I, i.e., ℂA ∖ ℝA splits in two connected components, determine the Euler characteristic of these components.

(4) Estimate from above the number of ovals which may have such a curve ℂA.