Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).
These might be used as a supplement in "Introduction to Modern Geometry" (MATH 4157/5157). The catalog description of this course (as of fall 2021) is: "An introduction to Euclidean and non-Euclidean geometries, emphasizing the distinction between the axiomatic characterizations, and the transformational characterizations of these geometries. Some history of the development of the discipline will also be included." This is the only undergraduate geometry class at ETSU.
- Chapter 1. Affine and Projective Algebraic Sets.
- Chapter 2. Basic Notions of Elimination Theory and Applications.
- Chapter 3. Zariski Closed Subsets and Ideals in the Polynomials Ring.
- Chapter 4. Some Topological Properties.
- Chapter 5. Regular and Rational Functions.
- Chapter 6. Morphisms.
- Chapter 7. Rational Maps.
- Chapter 8. Product of Varieties.
- Chapter 9. More on Elimination Theory.
- Chapter 10. Finite Morphisms.
- Chapter 11. Dimension.
- Chapter 12. The Cayley Form.
- Chapter 13. Grassmannians.
- Chapter 14. Smooth and Singular Points.
- Chapter 15. Power Series.
- Chapter 16. Affine Plane Curves.
- Chapter 17. Projective Plane Curves.
- Chapter 18. Resolution of Singularities of Curves.
- Chapter 19. Divisors, Linear Equivalence, Linear Series.
- Chapter 20. The Riemann-Roch Theorem.
Chapter 1. Affine and Projective Algebraic Sets.
- Section 1.1. Affine Algebraic Sets.
- Section 1.2. Projective Spaces.
- Section 1.3. Graded Rings.
- Section 1.4. Projective Algebraic Sets.
- Section 1.5. Projective Closure of Affine Sets.
- Section 1.6. Examples.
- Section 1.7. Solutions of Some Exercises.
Chapter 2. Basic Notions of Elimination Theory and Applications.
- Section 2.1. The Resultant of Two Polynomials.
- Section 2.2. The Intersection of Two Plane Curves.
- Section 2.3. Kronecker Elimination Method: One Variable.
- Section 2.4. Kronecker Elimination Method: More Variables.
- Section 2.5. Hilbert Nullstellensatz.
- Section 2.6. Solutions of Some Exercises.
Chapter 3. Zariski Closed Subsets and Ideals in the Polynomials Ring.
- Section 3.1. Ideals and Coordinate Rings.
- Section 3.2. Examples.
- Section 3.3. Solutions of Some Exercises.
Chapter 4. Some Topological Properties.
- Section 4.1. Irreducible Sets.
- Section 4.2. Noetherian Spaces.
- Section 4.3. Topological Dimension.
- Section 4.4. Solutions of Some Exercises.
Chapter 5. Regular and Rational Functions.
- Section 5.1. Regular Functions.
- Section 5.2. Rational Functions.
- Section 5.3. Local Rings.
- Section 5.4. Integral Elements over a Ring.
- Section 5.5. Subvarieties and Their Local Rings.
- Section 5.6. Product of Affine Varieties.
- Section 5.7. Solutions of Some Exercises.
Chapter 6. Morphisms.
- Section 6.1. The Definition of Morphism.
- Section 6.2. Which Maps are Morphisms.
- Section 6.3. Affine Varieties.
- Section 6.4. The Veronese Morphism.
- Section 6.5. Solutions of Some Exercises.
Chapter 7. Rational Maps.
- Section 7.1. Definition of Rational Maps and Basic Properties.
- Section 7.2. Birational Models of Quasi-projective Varieties.
- Section 7.3. Unirational and Rantional Varieties.
- Section 7.4. Solutions of Some Exercises.
Chapter 8. Morphisms.
- Section 8.1. Segre Varieties.
- Section 8.2. Products.
- Section 8.3. The Blow-up.
- Section 8.4. Solutions of Some Exercises.
Chapter 9. More on Elimination Theory.
- Section 9.1. The Fundamental Theorem of Elimination Theory.
- Section 9.2. Morphisms on Projective Varieties Are Closed.
- Section 9.3. Solutions of Some Exercises.
Chapter 10. Finite Morphisms.
- Section 10.1. Definitions and Basic Results.
- Section 10.2. Projections and Noether's Normalization Theorem.
- Section 10.3. Normal Varieties and Normalization.
- Section 10.4. Ramification.
- Section 10.5. Solutions of Some Exercises.
Chapter 11. Dimension.
- Section 11.1. Characterization of Hypersurfaces.
- Section 11.2. Intersection with Hypersurfaces.
- Section 11.3. Morphisms and Dimension.
- Section 11.4. Elimination Theory Again.
- Section 11.5. Solutions of Some Exercises.
Chapter 12. The Cayley Form.
- Section 12.1. Definition of the Cayley Form.
- Section 12.2. The Degree of a Variety.
- Section 12.3. The Cayley Form and Equations of a Variety.
- Section 12.4. Cycles and Their Cayley Forms.
- Section 12.5. Solutions of Some Exercises.
Chapter 13. Grassmannians.
- Section 13.1. Plücker Coordinates.
- Section 13.2. Grassmann Varieties.
- Section 13.3. Solutions of Some Exercises.
Chapter 14. Smooth and Singular Points.
- Section 14.1. Basic Definitions.
- Section 14.2. Some Properties of Smooth Points.
- Section 14.3. Smooth Curves and Finite Maps.
- Section 14.4. A Criterion for a Map to Be an Isomorphism.
- Section 14.5. Solutions of Some Exercises.
Chapter 15. Power Series.
- Section 15.1. Formal Power Series.
- Section 15.2. Congruences, Substitutions and Derivatives.
- Section 15.3. Fractional Power Series.
- Section 15.4. Solutions of Some Exercises.
Chapter 16. Affine Plane Curves.
- Section 16.1. Multiple Points and Principal Tangent Lines.
- Section 16.2. Parametrizations and Branches of a Curve.
- Section 16.3. Intersections of Affine Curves.
- Section 16.4. Solutions of Some Exercises.
Chapter 17. Projective Plane Curves.
- Section 17.1. Some Generalities.
- Section 17.2. M. Noether's Af + Bg Theorem.
- Section 17.3. Applications of the Af + Bg Theorem.
- Section 17.4. Solutions of Some Exercises.
Chapter 18. Resolution of Singularities of Curves.
- Section 18.1. The Case of Ordinary Singularities.
- Section 18.2. Reduction to Ordinary Singularities.
Chapter 19. Divisors, Linear Equivalence, Linear Series.
- Section 19.1. Divisors.
- Section 19.2. Linear Equivalence.
- Section 19.3. Fibres of a Morphism.
- Section 19.4. Linear Series.
- Section 19.5. Linear Series and Projective Morphisms.
- Section 19.6. Adjoint Curves.
- Section 19.7. Linear Sytems of Plane Curves and Linear Series.
- Section 19.8. Solutions of Some Exercises.
Chapter 20. The Riemann-Roch Theorem.
- Section 20.1. The Riemann-Roch Theorem.
- Section 20.2. Consequences of the Riemann-Roch Theorem.
- Section 20.3. Differentials.
- Section 20.4. Solutions of Some Exercises.
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