Algebraic Geometry

Year
1
Academic year
2023-2024
Code
02002056
Subject Area
Mathematics
Language of Instruction
Portuguese
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Elective
Level
2nd Cycle Studies - Mestrado

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 in an accessible way to the students taking the course.

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

(1st Part - Affine Varieties) Zariski toplogy on affine space. Hilbert's basis theorem. Vanishing ideal of an affine variety. Hilbert's Nullstellensatz. Maximal spectrum of a ring. Irreducibility and dimension. Morphisms of affine varieties. Finite morphisms.

(2nd Part - Algebraic Varieties) Category of pre-varieties. Sheaves of regular functions. Induced and quotient structures. Projective space and quasi-projective varieties. Algebraic varieties. Products of algebraic varietites. Segre embedding. Topology of algebraic varieties.

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

J. Neves, Introdução à  Geometria Algébrica, Departamento de Matemática da FCTUC, 2012.
M. Reid, Undergraduate Algebraic Geometry, LMS Student Texts, Cambridge University Press, 1989.
I. Shafarevich, Basic Algebraic Geometry 1, Springer-Verlag, 1994.
J. Harris, Algebraic Geometry. A First Course, Graduate Texts in Mathematics, 133, Springer-Verlag, 1985.
G. Kempf, Algebraic Varieties, London Mathematical Society Lecture Note Series 172, Cambridge University Press, 1993.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977.