Theory of schemes and stacks, Derived Categories and Intersection theory II
This course is targeting 3 major segments of algebraic geometry. The theory of schemes and stacks following Grothendieck’s EGA / SGA and Deligne’s work on stacks. The derived category of coherent sheaves, its properties and relevance to birational geometry and moduli theory, and intersection theory on schemes and stacks. Students who passed a basic course in algebraic geometry (e.g. Hartshorne’s first 3 chapters) must be able to follow the course with no further pre-requisite. The course will offer an extensive analysis of all three segments and aims at providing the detailed background for algebraic geometry students to perform advanced research on geometry of schemes and moduli, say in the context of birational geometry, enumerative algebraic geometry or derived algebraic geometry.
Lecturer
Date
21st February ~ 20th June, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 09:50 - 12:15 | Shuangqing-B725 | ZOOM 09 | 230 432 7880 | BIMSA |
Prerequisite
Basic course in Algebraic Geometry (first 3 chapters of Hartshorne)
Syllabus
1. Review of Scheme theory
• 1.1: Review of Schemes (Hartshorne or EGA I, II)
• 1.2. Review of Sheaf Cohomology (EGA 3)
• 1.3. Residue and Duality (EGA 10)
2. Theory of stacks
• 2.1 Champs Algebrique (G. Lamoun)
• 2.2. Sheaf Cohomology, Lefschetz Local to global theorems (SGA 2)
3. Intersection theory
• 3.1 Intersection theory and Grothendieck -Riemann-Roch (EGA 11, SGA 6)
• 3.2 Fulton’s deformation to normal cone (Fulton’s Intersection theory)
• 3.3. Computations (Fulton’s Intersection theory)
4. Derived categories of coherent sheaves:
4.1. Derived Category of coherent sheaves, Properties (FM transforms in AG, Daniel Huybrechts) 4.2 Derived Category and canonical bundle (FM transforms in AG, Daniel Huybrechts)
4.3. FM transforms (FM transforms in AG, Daniel Huybrechts)
4.4. Spherical and Exceptional objects (FM transforms in AG, Daniel Huybrechts)
5. Let us now compute! Intersection theory in derived category:
5.1. Counting Curves in derived category (Fulton’s intersect. theory and collective papers in DT/GW) 5.2. Counting surfaces in derived category (Fulton’s intersect. theory and collective papers in DT/GW)
Note: This is the collective syllabus for both Fall 2024 and Spring 2025 courses combined.
• 1.1: Review of Schemes (Hartshorne or EGA I, II)
• 1.2. Review of Sheaf Cohomology (EGA 3)
• 1.3. Residue and Duality (EGA 10)
2. Theory of stacks
• 2.1 Champs Algebrique (G. Lamoun)
• 2.2. Sheaf Cohomology, Lefschetz Local to global theorems (SGA 2)
3. Intersection theory
• 3.1 Intersection theory and Grothendieck -Riemann-Roch (EGA 11, SGA 6)
• 3.2 Fulton’s deformation to normal cone (Fulton’s Intersection theory)
• 3.3. Computations (Fulton’s Intersection theory)
4. Derived categories of coherent sheaves:
4.1. Derived Category of coherent sheaves, Properties (FM transforms in AG, Daniel Huybrechts) 4.2 Derived Category and canonical bundle (FM transforms in AG, Daniel Huybrechts)
4.3. FM transforms (FM transforms in AG, Daniel Huybrechts)
4.4. Spherical and Exceptional objects (FM transforms in AG, Daniel Huybrechts)
5. Let us now compute! Intersection theory in derived category:
5.1. Counting Curves in derived category (Fulton’s intersect. theory and collective papers in DT/GW) 5.2. Counting surfaces in derived category (Fulton’s intersect. theory and collective papers in DT/GW)
Note: This is the collective syllabus for both Fall 2024 and Spring 2025 courses combined.
Reference
Grothendieck EGA I,II, III, X, XI
Grothendieck SGA VI
Champs Algebrique (G. Lamoun)
Fourier Mukai Transform in Algebraic Geometry (Daniel Huybrechts) Intersection theory (W. Fulton)
Grothendieck SGA VI
Champs Algebrique (G. Lamoun)
Fourier Mukai Transform in Algebraic Geometry (Daniel Huybrechts) Intersection theory (W. Fulton)
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He joined BIMSA as a Professor in September 2023. Prior to BIMSA he was a senior personnel (Professor) at Simons Collaboration Program on Homological Mirror Symmetry at Harvard University Center for Mathematical Sciences and Applications (CMSA) for 7 years, during which he was also an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). He is working on geometry of moduli spaces of sheaves and curves from enumerative geometry point of view as well as studying their structural properties from derived geometry and geometric representation theory point of view. He has been an invited speaker to ICCM and ICBS and a recipient of several awards. In 2019 he was one of the 30 winners of the IRFD "Research Leader" grant (approx 1M USD) in 2019.