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Course, academic year 2023/2024
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Algebraic Curves - NMAG302
Title: Algebraické křivky
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Additional information: https://www2.karlin.mff.cuni.cz/~stanovsk/vyuka/krivky.htm
Guarantor: doc. RNDr. David Stanovský, Ph.D.
Class: M Bc. MMIB
M Bc. MMIB > Povinné
M Bc. MMIT
M Bc. MMIT > Povinné
M Bc. OM
M Bc. OM > Zaměření MSTR
M Bc. OM > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NMIB054
Interchangeability : NMIB054
Is interchangeable with: NMIB054
In complex pre-requisite: NMAG349
Annotation -
A recommended course for Information Security and specialization Mathematical Structures within General Mathematics. This is an introductory lecture to basic algebraic geometry focused on curves. The course is concerned with the basic notions (affine and projective variety, mappings on varieties, coordinate rings), local properties of curves, Bezout theorem and elliptic curves.
Last update: G_M (15.05.2012)
Course completion requirements -

The credit (zápočet) will be granted with the exam.

The exam is written, containing both theoretical and computational problems, based on the topics covers by the lecture and exercise sessions.

See the course website for details.

Last update: Stanovský David, doc. RNDr., Ph.D. (22.02.2022)
Literature -

W. Fulton: Algebraic Curves: an introduction to algebraic geometry, Benjamin, Reading 1969.

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

J. H. Silverman and J. Tate: Rational Points on Elliptic Curves, Springer, New York 1992.

I. R. Shafarevich: Basic Algebraic Geometry 1, Springer, Berlin 1994.

Last update: G_M (24.04.2012)
Requirements to the exam -

The topics covered by the exam correspond to the topics presented at the lecture and the exercise sessions, see http://www.karlin.mff.cuni.cz/~stanovsk/vyuka/krivky.htm

Last update: Stanovský David, doc. RNDr., Ph.D. (18.02.2020)
Syllabus -

Algebraic geometry in affine spaces

  • Galois correspondence IV, Hilbert's Nullstellensatz
  • irreducible decomposition
  • coordinate rings, local properties of curves

Algebraic geometry in projective spaces

  • projective specas, homogeneous polynomials and ideals
  • projective irreducibility, projective Nullstellensatz
  • relation of affine and projective curves
  • Bezout's theorem
  • elliptic curves.
Last update: Stanovský David, doc. RNDr., Ph.D. (20.02.2018)
Entry requirements -

Some familiarity with basics of commutative algebra, properties of polynomial rings over a field and algebraic varieties.

Last update: G_M (24.04.2012)
 
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