FSF3606 Algebraic Geometry III 7.5 credits

• 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

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.

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.

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

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

• 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.

Approved homework assignments, and lectures.

David Rydh

• 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.