ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
Provide the students with a basic formation in Algebraic Geometry, which will allow them to understand the problems it deals with.
Illustrate geometrically basic tools and results of Commutative Algebra.
Develop the geometry of the sets of solutions of systems of algebraic equations in the affine space and of systems of algebraic equations in the projective space. We will introduce the notion of sheaf of regular functions on a topological space to unify both points of view and several areas of Algebra and Geometry.
Understand the usefulness of the structure of algebraic variety.
1 Algebraic sets and regular functions. (5 sessions)
1.1 Affine algebraic sets. The Zariski topology.
1.2 Polynomial maps and coordinate rings.
1.3 Examples of affine algebraic sets and coordinate rings.
1.4 Affine varieties. Regular functions.
2. Rational functions and morphisms. (5 sessions)
2.1 Sheaves of functions.
2.2 Structure sheaf of an affine algebraic set.
2.3 Local ring at a point and at a subvariety.
2.4 Regular morphisms.
2.5 Products of algebraic sets.
3. Varieties. (5 sessions)
3.1 Prevarieties. The field of rational functions.
3.2 Hausdorff’s axiom: separateness.
3.3 Varieties. Dominant morphisms. Birational equivalence.
3.4 Rational maps between varieties.
4. Local study. Nonsingular varieties. (5 sessions)
4.1. Local study of curves.
4.2 Tangent spaces and singularity. Tangent cone and tangent space.
4.3 Nonsingular points.
4.4 Nonsingularity and differentials.
4.5 Normal varieties and morphisms.
5. Projective varieties. (5 sessions)
5.1 Algebraic structure of the n-dimensional projective space.
5.2 The projective space is a variety.
5.4 Complete algebraic spaces.
5.3 Constructions and examples of morphisms.
5.4 Graßmann varieties.
6. Dimension. (3 sessions)
6.1 Topological description of dimension.
6.2 Dimension as transcendental degree.
6.3 Intersections in the affine and projective space.
6.4 Finite morphisms. Nöther’s normalization theorem.
6.5 Fibers. Morphisms and dimension.
Basic bibliography
Gathman, A.: Algebraic Geometry, notes Technische Universität Kaiserslautern, 2022, https://www.mathematik.uni- kl.de/~gathmann/en/alggeom-2021.php
Kempf, G. R.: Algebraic Varieties, London Math. Soc. Lecture Notes Series, 172, Cambridge University Press, Cambridge, 1993.
Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985.
J.S. Milne: Algebraic Geometry, disponible en: http://www.jmilne.org/math/CourseNotes/ag.html
Complementary bibliography
Bump, Daniel.: Algebraic geometry. World Scientific Publishing Co., Inc., River Edge, Nueva Jersey, 1998.
Dieudonné, J.: Cours de géométrie algébrique, 2/ Précis de géométrie algébrique élémentaire, Presses Univ. France, París, 1974.
Dieudonné, J.: History of Algebraic Geometry, Wadsworth Advanced Books & Software, Monterrey (California), 1985.
Fulton, W.: Algebraic Curves, W. A. Benjamin, Nueva York, 1969.
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Math. 52, Springer–Verlag, Heidelberg, 1977
Hulek, K.: Elementary Algebraic Gaometry, Student Mathematical Library, Volume 20, AMS, 2003.
Mumford, D.: The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer–Verlag, Heidelberg, 1988.
Perrin, D.: Algebraic geometry. An introduction. Springer-Verlag London, 2008
CG01 - Introduce students into the research, as an integral part of a deep formation, preparing them for the eventual completion of a doctoral thesis.
CG02 - Acquisition of high level mathematical tools for diverse applications covering the expectations of graduates in mathematics and other basic sciences.
CG03 - Know the broad panorama of current mathematics, both in its lines of research, as well as in methodologies, resources and problems it addresses in various fields
CB6 - Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context
CB10 - That students have the learning skills that allow them to continue studying in a way that will be largely self-directed or autonomous.
CE01 - Train for the study and research in mathematical theories in development.
Lectures by the teacher and some expositions in class of the program by the students.
There will be also some sessions devoted to the resolution of problems.
- Class participation (10%)
- Completion of proposed exercises and presentation of their solutions in class (20%)
- Oral or written final exam, depending on the number of students (70%)
By regulation: "The qualification of the student will not be inferior to the final exam nor to the obtained weighting it with the continuous evaluation, giving to this last one a weight not inferior to 25%".
About 6 hours of work per week (75 hours per term, including 2 to 3 lectures per week).
Previous knowledge of Algebra, especially Commutative Algebra, is necessary. The knowledge taught in the course Commutative Algebra of this master's degree is sufficient.
In cases of fraudulent performance of exercises or tests, the provisions of the Regulations on the Evaluation of Students' Academic Performance and the Review of Grades will apply.
Leovigildo Alonso Tarrio
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813159
- leo.alonso [at] usc.es
- Category
- Professor: University Lecturer
| Wednesday | |||
|---|---|---|---|
| 13:00-14:00 | Grupo /CLE_01 | Spanish | Classroom 10 |
| Thursday | |||
| 13:00-14:00 | Grupo /CLIL_01 | Spanish | Classroom 10 |
| 05.28.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 07.07.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |