Algebraic Curves - NMAG302
Title in English: Algebraické křivky
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. David Stanovský, Ph.D.
Class: M Bc. MMIB
M Bc. MMIB > Povinné
M Bc. MMIT > Povinné
M Bc. OM
M Bc. OM > Zaměření MSTR
M Bc. OM > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NMIB054
Interchangeability : NMIB054
In complex pre-requisite: NMAG349
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Annotation -
Last update: G_M (15.05.2012)
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.
Course completion requirements -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (19.02.2019)

The credit will be granted for homework solutions. There will be 3 sets of problems. The necessary condition is at least 50 % of points. If the conditions are not met, it is still possible to get the credit under upon solving additional problems. The extent and deadlines for these will be decided by the lecturer.

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

Literature -
Last update: G_M (24.04.2012)

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.

Requirements to the exam -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (19.02.2019)

The topics covered by the exam correspond to the topics presented at the lecture and the exercise sessions.

Syllabus -
Last update: doc. RNDr. David Stanovský, Ph.D. (20.02.2018)

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.
Entry requirements -
Last update: G_M (24.04.2012)

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