Teaching plan for the course unit



Close imatge de maquetació




General information


Course unit name: Introduction to Algebraic Geometry

Course unit code: 574267

Academic year: 2021-2022

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



-  Lecture





-  Problem-solving class




Supervised project


Independent learning






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 contents
are worked upon through problem-solving sessions. Student have to solve problems in the
blackboard during these sessions. Occasionally, during the problem-solving sessions
complementary 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 or
submitted for correction), worth between 30% and 50%;
— paper and oral presentation on a topic related to the content of the subject: worth
between 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

Consulteu la disponibilitat a CERCABIB


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,

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.