- Oggetto:
Topics in Algebraic Geometry
- Oggetto:
Topics in Algebraic Geometry
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Academic year 2023/2024
- Teacher
- Riccardo Moschetti
- Teaching period
- Jan-Mar
- Type
- Basic
- Credits/Recognition
- 5
- Course disciplinary sector (SSD)
- MAT/03 - geometry
- Delivery
- Formal authority
- Language
- English
- Attendance
- Obligatory
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Sommario del corso
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Program
Introduction to derived categories and their use in algebra and geometry.
Derived categories have become a very powerful tool in algebraic geometry. We aim to give an introduction to this language, giving equal importance to the theoretical background and to the presentation of research problems in which derived categories play a leading role.
Hours: 25
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Course delivery
Here is the proposed course schedule. During the course this could be updated in case of necessities of the participants.
19/2/2024, 15:30 - 18:30 - Sala S
Introduction to the topics of the course, and an overview of category theory.
04/3/2024, 14:30 - 17:30 - Sala S
Axioms for the definition of triangulated categories. Interpretation with respect to the structure of an abelian category. Exact functors, Serre functors, and their connection with adjoint functors and equivalences. Semiorthogonal decompositions and exceptional objects, examples drawn from algebraic geometry.
11/3/2024, 14:30 - 17:30 - Sala S
Construction of the derived category of an abelian variety with its triangulated category structure. Hints at the proof that the derived category of a smooth connected projective variety with trivial canonical bundle does not have semiorthogonal decompositions.
18/3/2024, 14:30 - 17:30 - Sala S
Definition of derived functor, examples in the context of geometry. Derived category of the double point.08/4/2024, 14:30 - 17:30 - Sala S
Fourier-Mukai functors and their properties.
15/04/2024, 14:30 - 17:30 - Sala S
22/04/2024, 14:30 - 17:30 - Sala S
29/04/2024, 14:30 - 18:30 - Sala S
Suggested readings and bibliography
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Andrei Căldăraru, Derived categories of twisted sheaves on Calabi-Yau manifolds (link)
Andrei Căldăraru, Derived categories of sheaves: a skimming (link)
Daniel Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry (link)
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