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Proseminar on Algebraic Plane Curves

​Algebraic Geometry is a relevant branch of mathematics with a large number of sinergies with other fields. Broadly, Algebraic Geometry studies the set of solutions of a system of polynomial equations, which are called algebraic varieties. This proseminar serves as a first contact with the world of algebraic geometry. We will focus on a very intuitive but still challenging family of algebraic varieties: algebraic plane curves. These are the set of solutions of a polynomial in two variables. For instance, circles, parabolas and hyperbolas are examples of algebraic plane curves. In this proseminar we will analyse the geometry of this objects from an algebraic approach.

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The requirements for this course are linear algebra and some knowledge on rings and fields. The proseminar will take place on Thursdays 10-12 am at the room S08. The first lecture will be on Thursday 25th

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We will mainly follow the book:

  • Gibson, C. G. (1998). Elementary geometry of algebraic curves: an undergraduate introduction. Cambridge University Press.

Seminar Talks

​All the sections mentioned in each talk correspond to Gibson book (see above)

  1. ​Algebraic plane curves: definitions and first examples. Contents: Sections 1.1, 1.2, 2.1, 2.3, 2.4. (not available)

  2. Polynomial algebras. Contents: Chapter 3 (not available).

  3. Affine equivalence and affine conics: Contents: Sections 4.1, 4.2, 4.3, 5.1. (not available)

  4. Singularities. Contents: Chapter 6.

  5. Projective curves. Contents: Chapter 9. (not available)

  6. Singular projective curves. Contents: Chapter 10.

  7. Projective equivalence. Contents: Chapter 11.

  8. Tangents to affine and projective curves. Contents: Chapter 7 and Sections 12.1, 12.2.

  9. Resultants. Contents: Sections 14.1, 14.2, 14.3.

  10. Bèzout's Theorem. Contents: Sections 14.4, 14.5, 14.6.

  11. Projective cubics. Contents: Chapter 15.

  12. Linear systems. Contents: Chapter 16.

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