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FSF3606 Algebraic Geometry III 7.5 credits

Administer About course

Course offerings are missing for current or upcoming semesters.
Headings with content from the Course syllabus FSF3606 (Spring 2019–) are denoted with an asterisk ( )

Content and learning outcomes

Course contents

  • Čech cohomology of schemes

  • Cohomology of line bundles on projective space

  • Finiteness of cohomology (projective and proper schemes)

  • Serre vanishing

  • Serre’s cohomological criterion for  ampleness

  • Riemann–Roch theorem (for non-singular  curves)

  • Serre duality (for smooth varieties)

  • Hilbert polynomials of coherent sheaves

  • Higher direct images

  • Some results on rational and elliptic curves

  • Riemann–Hurwitz formula

  • Derived functors (Tor, Ext and sheaf cohomology)

  • Cohomology and base change

Intended learning outcomes

After the course, the student should have sufficient knowledge of the basic principles in algebraic geometry to be able to study more advanced topics as well as some research articles in algebraic geometry.

Literature and preparations

Specific prerequisites

A Master degree including at least 30 university credits (hp) in in Mathematics.

Knowledge of basic algebraic geometry (schemes, sheaves, etc.) on the level of Algebraic Geometry II (FSF3605). This implies that a solid basic knowledge of topology and commutative algebra is needed, for instance SF2735 Homological Algebra and Algebraic Topology and SF2737 Commutative Algebra and Algebraic Geometry. To have taken a second course in commutative algebra is also desirable.

Recommended prerequisites

No information inserted

Equipment

No information inserted

Literature

R. Vakil, Foundations of Algebraic Geometry (Math 216) och R. Hartshorne, Algebraic Geometry.

Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

Grading scale

G

Examination

  • ÖVN1 - Exercises, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Each participant should give two one-hour lectures. They should also discuss the material with a teacher before giving the lecture. In addition, there are homework problems to be handed in.

Other requirements for final grade

Approved homework assignments, and lectures.

Opportunity to complete the requirements via supplementary examination

No information inserted

Opportunity to raise an approved grade via renewed examination

No information inserted

Examiner

Profile picture David Rydh

Ethical approach

  • All members of a group are responsible for the group's work.
  • In any assessment, every student shall honestly disclose any help received and sources used.
  • In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

Further information

Course room in Canvas

Registered students find further information about the implementation of the course in the course room in Canvas. A link to the course room can be found under the tab Studies in the Personal menu at the start of the course.

Offered by

SCI/Mathematics

Main field of study

This course does not belong to any Main field of study.

Education cycle

Third cycle

Add-on studies

No information inserted

Contact

David Rydh (dary@kth.se)

Postgraduate course

Postgraduate courses at SCI/Mathematics