Unit ALGEBRAIC GEOMETRY

Course
Mathematics
Study-unit Code
55A00044
Curriculum
Matematica per la crittografia
Teacher
Alessandro Tancredi
Teachers
  • Alessandro Tancredi
Hours
  • 63 ore - Alessandro Tancredi
CFU
6
Course Regulation
Coorte 2021
Offered
2022/23
Learning activities
Caratterizzante
Area
Formazione teorica avanzata
Academic discipline
MAT/03
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
Algebraic varieties as ringed spaces. Affine and projective varieties. Dimension of an algebraic variety. Regular and singular points of an algebraic variety.
Reference texts
J. Bochnak, M. Coste, M. F. Roy, Real algebraic geometry. Springer 1998
D. Munford, The red book of varieties and schemes. Springer 1988
I. R. Shafarevich, Basic Algebraic Geometry. Springer 1974

Further notes and references will be supplied by the lecturer
Educational objectives
The course introduces to the theory of algebraic varieties as ringed spaces. Its goal is to familiarize the students with the tools they will need in order to use algebraic varieties, also with regard to other fields of the geometry.
Prerequisites
Elements of commutative algebra and fields theory, wich are stated as needed, and some elementary topology.
Teaching methods
face-to-face, office hours, usage of the platform “Unistudium” (https://www.unistudium.unipg.it)
Other information

Learning verification modality
The final exam consists in an oral discussion of about an hour on the subjects developped during the course. A detailed list of the subjects is provided at the end of the lectures. The aim of the exam is to evaluate the level and the quality of the knowledge the students have acquired and to check their ability in the exposition.
Extended program
Noetherian topological spaces. Sheaves and ringed spaces. Algebraic sets. Zariski topology. Polynomial and regular functions on algebraic sets. Affine varieties. Prevarieties and their morphisms: products of prevarieties. Algebraic varieties. Rational morphisms. Dimension of a variety. The local ring of a point of an algebraic variety: tangent and cotangent spaces. Regular and singular points of an algebraic variety. Algebraic varieties over an algebraic closed and over a really closed field. Algebraic transversality. Smooth morphisms of algebraic varieties. Projective and complete varieties. Fibre of a morphism. Finite morphisms. Bertini’s theorems. Complexification af an affine and projective real agebraic set. Analytic structure of real and complex varieties.
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