Teaching plan for the course unit

 

 

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General information

 

Course unit name: Local and Global Theory of Algebraic Curves

Course unit code: 364205

Academic year: 2021-2022

Coordinator: Rosa Maria Miro Roig

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

Face-to-face

 

30

 

-  Lecture with practical component

Face-to-face

 

30

Supervised project

20

Independent learning

70

 

 

Recommendations

 

It is advisable to have basic knowledge of projective geometry and be familiar with the basic concepts of algebra.

 

 

Competences to be gained during study

 

   -

Capacity to study the intrinsic and extrinsic geometry of an algebraic curve.

 

 

Learning objectives

 

Referring to knowledge

— Acquire basic knowledge of the local, projective and intrinsic theories of algebraic curves.

 

Referring to abilities, skills

— Perform introductory mathematical research in this subject area.

— Utilise basic bibliographic reference material on the subject.

 

 

Teaching blocks

 

1. Implicit equations of plane curves; Intersection of plane curves (resultant)

2. Linear systems of curves

3. Parametrisation of curves

4. Local properties of curves

5. Bézout’s theorem and its applications

6. Plücker formulas; Applications

7. Rational curves; Curves of small genus

8. Higher-dimensional varieties

 

 

Teaching methods and general organization

 

• In the expected blended teaching model:

The course is distributed as follows: 2 hours of lectures per week, 2 hours of problem-solving sessions per week and a tutored assignment.

Lectures will be online and asynchronous. Class notes will be distributed every week with the main definitions, demonstrations and results, and examples to illustrate them. Complementary materials will also be distributed.

Problem-solving sessions will be carried out face-to-face. The first part of each session will be devoted to the review of the class notes distributed in advance. The rest of the class will consist of solving problems/exercises previously assigned.

The tutored assignment must be completed throughout the course. Follow-up tutoring session will be scheduled by appointment.

• In the event that virtual teaching is required due to the health situation:

Both lectures and problem-solving sessions will be carried out completely in distance learning mode.

• In the event of face-to-face teaching:

Two hours of lectures per week, two hours per week dedicated to problem-solving exercises and a supervised project structured around complementary topics to stimulate independent learning.


In all three scenarios (face-to-face, blended or virtual learning), lectures are used to provide the main definitions and results related to the subject, and to illustrate these with examples. A series of exercises must then be completed for each topic, and solutions presented in class or submitted via the Virtual Campus. Over the semester a project related to the content of the course will be assigned, to be completed individually (or in groups) and presented in class (or virtually) at some point during the course, or in a dedicated session scheduled after the theory classes have finished.

 

 

Official assessment of learning outcomes

 

Continuous assessment consists of:
— Weekly problem-solving exercises to be presented orally in face-to-face sessions: 20%
— Mid-term assessed activity: 30%
— Tutored assignment and oral presentation: 15%
— Final examination: 35%

Depending on the health situation, assessed activities may be: face-to-face tests, online synchronous tests or assignments to be submitted.

A minimum mark of 3.5 in the final examination is required to be eligible to pass the subject.

Repeat assessment

Students with a final grade of 3 or higher are entitled to repeat assessment. The final grade for the subject is then the grade for the repeat assessment exam.

This examination may be carried out face-to-face or online, depending on the health situation.

 

Examination-based assessment

Students following single assessment take an examination on the whole contents of the subject, on the same date as the final examination for continuous assessment. This option must be requested of the Secretary’s Office at the Faculty before the established date. The mark obtained in this examination is the final grade for the subject.

This examination may be carried out face-to-face or online, depending on the health situation.

Repeat assessment

Students with a final grade of 3 or higher are entitled to repeat assessment. The final grade for the subject is then the grade for the repeat assessment exam.

This examination may be carried out face-to-face or online, depending on the health situation.

 

 

Reading and study resources

Consulteu la disponibilitat a CERCABIB

Book

Fulton, W. Curvas algebraicas : introducción a la geometría algebraica. Barcelona [etc.] : Reverté, 2005.  EnllaƧ

 

Griffiths, P. Introduction to algebraic curves. Providence (R.I.) : American Mathematical Society, 1989.  EnllaƧ

Miranda, R. ALgebraic curves and Riemann surfaces, Graduate Studies in Mathematics. Providence (R.I.) : American Mathematical Society, 1995.  EnllaƧ

 

M. Reid, Undergraduate Algebraic Geometry, LMS Students Texts 12, Cambridge Univ. Press, 1988

R.J. Walker, Algebraic curves, Springer-Verlag, 1978

K. Hulek, Elementary Algebraic Geometry, Student Mathematical Library AMS
Volume: 20; 2003;