Algebraic Geometry

Recommended Prerequisites

Basic courses in Linear Algebra, Algebra, Geometry and Topology.

Teaching Methods

Classes are expository and include examples and exercises sessions, where students are expected  to explain to the rest of the group their solutions to the exercises.

During the semester students may use tutorial time to clarify their difficulties in grasping the theory.

Learning Outcomes

The main goal of the course is to make the student acquainted with the language, concepts and techniques of Algebraic Geometry. The first half of the course covers part of the classical fundamental theory, including the necessary requisites of Commutative Algebra. In the second half, the course is intended to make an introduction to the advanced fundamental concepts of Algebraic Geometry.

The course aims at developing the following skills: knowledge of mathematical results; ability to formulate and solve problems and, more precisely, the ability to relate distinct areas of Mathematics. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

Affine varieties. Zariski's topology. Hilbert's Basis Theorem and Nullstellensatz. Spectrum of a ring. Projective varieties. Sheaves, schemes and algebraic varieties. Irreducibility and dimension. Morphisms of algebraic varieties.

One or more of the following additional topics may be covered:

Algebraic curves, Serre duality, the Riemann-Roch Theorem, sheaf cohomology.

Head Lecturer(s)

Jorge Manuel Sentieiro Neves

Assessment Methods

Assessment
There are 2 types of assessment: by mid-term exams or by a final examination: 100.0%

Bibliography

R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York-Heidelberg, 1977.

B. Hassett, Introduction to algebraic geometry, Cambridge University Press, Cambridge, 2007.

M. Reid, Undergraduate Algebraic Geometry, LMS Student Texts, vol. 12, Cambridge University Press, Cambridge, 1988.

I.R. Shafarevich, Basic Algebraic Geometry, volumes 1 & 2, 3rd edition, Springer, Heidelberg, 2013.