Teaching plan for the course unit

 General information

Course unit name: Introduction to Algebraic Geometry

Course unit code: 574267

Coordinator: Juan Carlos Naranjo Del Val

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face and/or online activities 60
 -  Lecture Online 30 -  Problem-solving class Face-to-face 30
 Supervised project 30
 Independent learning 60

 Recommendations

 Projective Geometry and abstract Algebra

 Competences to be gained during study

 — Capacity to study the geometry of algebraic varieties defined over an algebraically closed field.

 Learning objectives
 Referring to knowledge Referring to knowledge — Acquire a basic knowledge of affine and projective algebraic varieties.— Understand basic algebraic structures that allow the construction and study of algebraic varieties.— Acquire the knowledge and capacities required to solve problems that help assimilate theory. Referring to abilities, skills— Be introduced to algebraic geometry.

 Teaching blocks

1. Affine and projective varieties

1.1. Affine and projective algebraic sets; Zariski topology, Noetherian spaces

1.2. Hilbert’s Nullstellensatz; Irreducibility

1.3. Regular functions and rational functions

1.4. Short introduction to sheaves

1.5. Morphisms; Quasi-projective varieties; rational maps and birational equivalence

1.6. Abstract algebraic varieties

2. Products of varieties and elimination theory

2.1. Products; Segre morphism

2.2. Elimination theorems

3. Dimension theory

3.1. Dimension as transcendence degree

3.2. Finite morphisms

3.3. Krull’s dimension; dimension of the fibers of a morphism

4. Grassmannians, Lines in surfaces of projective three-dimensional space

5. Local theory

5.1. Zariski’s tangent space and embeddings

5.2. Singular points; Jacobian  criterion; Singularities of hypersurfaces in the projective space

5.3. Normal varieties

6. Divisors and class groups

 Teaching methods and general organization

 During the theory sessions the contents of the teaching blocks are explained. These contentsare worked upon through problem-solving sessions. Student have to solve problems in theblackboard during these sessions. Occasionally, during the problem-solving sessionscomplementary aspects of the subject are covered if required.

 Official assessment of learning outcomes

 Students are assessed on the basis of the following activities:— assessment of work completed during the semester (problems solved in the blackboard orsubmitted for correction), worth between 30% and 50%;— paper and oral presentation on a topic related to the content of the subject: worthbetween 30% and 50%;— a final examination, worth until 40%. Repeat assessment:Repeat assessment consists of a written examination on the date established by the Faculty.    Examination-based assessment Single assessment consists of a written examination.

 Reading and study resources

Book

Harris, Joe. Algebraic Geometry : a first course. New York ; Springer, 1992.

Hartshorne, R. Algebraic geometry. New York : Springer, 2000.

Hasset, B. Introduction to Algebraic Geometry. Cambridge : Cambridge University, 2008.

Hulek, K. Elementary Algebraic Geometry. Providence [R.I.] : American Mathematical Society,
2003.

Mumford, D. The Red book of varieties and schemes, Berlin: Springer, 1999.

Shafarevich, I. R. Basic algebraic geometry, Berlin : Springer, cop. 1994
2nd ed.

Smith, K., Kahanpää, L., Kekäläinen, P., Traves,W., An Invitation to Algebraic Geometry,
Universitext, Springer Verlag, New York, 2000.