SYLLABUS FOR MATH 459

(A DETAILED VERSION)

Abbreviation for the reference books: (see the description of the course for the complete references)

KRV is the textbook by V.Kharlamov, V.Rokhlin, O.Viro;

R is the textbook of M.Reid;

W is the textbook of R.Walker; for example, a reference to KRV 0.1–0.4 means subsections from 1 to 4 in Chapter 0 (Introduction) of KRV

Week 1. Introduction: algebraic curves in ℝ^{2} and ℝP^{2}. Non-singular compact curves in ℝ^{2} of degree ≤ 4. Prohibitions via Bezout theorem. Harnack theorem and M-curves. Hilbert’s Problem 16. Reference: KRV 0.1–0.4, 0.10

Week 2. A brief survey of the basic topological notions. Homeomorphisms and topological properties (i.g., connectedness and simply connectedness). Homotopy and isotopy. Definition of surfaces (and manifolds). Orientation preserving diffeomorphisms. Topological circles in ℝ^{2}: Jordan Theorem, Peano curve. Ambient homeomorphisms and isotopy in ℝ^{n}. Examples: graphs and curves in ℝ^{2}, knots and surfaces in ℝ^{3}. Reference:

Week 3. ℝP^{2}, affine charts. Generalization: ℝP^{n} and its linear subspaces, projective transforms. Projective closure of a curve, changing charts. A curve and its locus. Reduced curves. Reference: KRV 1.1–1.2; R 1.4–1.6, 1.8; W 1.1

Week 4. ℂP^{2}, affine and projective complex curves, non-singular complex curve as an oriented closed surface (Riemann surface), action of the complex conjugation. Reference: KRV 0.6, 1.3–1.7; R 2.14–2.15

Homework 1

Week 5. Singular and ordinary points of real algebraic curves. A curve near its singular point, the link of a singularity, branches. Examples: nodes, cusps; the other types of simple singularities. The order of tangency, inflection points. Reference: KRV 0.9, 1.7–1.9; R 2.1, 2.12; W 3.2

Week 6. Theory of intersections for plane algebraic curves; Bezout theorem; intersection at a singular point. Reference: KRV 1.7, 1.11, 2.4.4; R 1.9–1.10; W 3.3

Midterm I

Week 7. The space of curves; the discriminant hypersurface. Rigid isotopies. Pencils and linear systems. Examples: pencils of conics and cubics. Reference: KRV 0.8, 2.1–2.4, R 1.11–1.13; W 3.4, 3.6

Week 8. Curves on orientable and non-orientable surfaces. Curves in ℝP^{2}. Intersection theory ( mod 2) Reference: KRV 2.4.1–2.4.3

Homework 2

Week 9. The Euler characteristic and genus of a closed (Riemann) surface; their properties (under cutting and pasting, etc.). Rigid isotopy and the topology of non-singular complex curve in ℂP^{2}. Reference:

Week 10. The Harnack theorem, M-curves, the simplest constructions by the method of a small variation. Curves of degree ≤ 5. Construction of M-curves by the methods of Harnack and Hilbert. Hilbert’s Problem 16. Classification of curves of degree 6 with respect to real isotopy and rigid isotopy. Reference: KRV 0.5, 2.4.5–2.4.7, 2.5

Midterm II

Week 11. The Ragsdale Conjecture, the Petrovskii inequality, the Rokhlin Congruence. The complex orientations and the Rokhlin formula. Reference: RKV 0.6, 0.11, 2.6

Week 12. Examples and phenomena related to the curves of degree ≤ 7. Reference: RKV 0.7

Homework 3

Week 13. Singularities revisited: the Milnor number, the Newton polygon, the modality. Bifurcations of the simplest real singularities. Diagrams of A’Campo– Gussein-Zade. Reference: RKV 1.10

Final