Academic Year/course:
2017/18
453 - Degree in Mathematics
27043 - Algebraic Curves
Syllabus Information
Academic Year:
2017/18
Subject:
27043 - Algebraic Curves
Faculty / School:
100 - Facultad de Ciencias
Degree:
453 - Degree in Mathematics
ECTS:
6.0
Year:
4
Semester:
First semester
Subject Type:
Optional
Module:
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5.1. Methodological overview
The general teaching methodology designed for this class is based on the following:
- Lectures
- Problem sessions and oral presentations
- Office hours
- Students’ individual work
5.2. Learning tasks
Las actividades de eprendizaje serán fundamentalmente, la asistencia a las clases teóricas, la participación en las presentaciones orales y la asistencia a tutorías y el trabajo personal (estudio y realización de ejercicios).
Learning activities will consist on: lectures, oral presentations, office hours and personal work (including homework assignments).
5.3. Syllabus
1. Algebraic Preliminaires
- Commutative rings and ideals.
- Rings of fractions.
- Polynomial rings. Homogeneous polynomials.
- Noetherian rings. The Hilbert basis theorem
2. Algebraic Varieties
- Affine algebraic sets and ideals of sets of points.
- Hilbert's nullstellensatz.
- Polynomial maps, Zariski's topology, morphisms and rational maps.
- The projective space. Projective algebraic sets.
- Varieties in a multiprojective space.
3. Algebraic Plane Curves}
- Parametrizable curves.
- Local properties: singularities, tangents and multiplicities.
- Multiplicities local rings.
- Bézout's theorem.
5.4. Course planning and calendar
Timetables, exams schedules and any other important information will be available through the Facultad de Ciencias web page.
5.5. Bibliography and recommended resources
E. BRIESKORN, H. KN\"ORRER. Plane Algebraic Curves (English edition). Springer, Basel 1986.
W. FULTON. Algebraic curves: An Introduction to Algebraic Geometry, 3rd Edition. Addison Wesley Publ. Co., Reading MA 2008.
F. KIRWAN. Complex algebraic curves. Cambridge University Press, Cambridge 1992.