FURTHER TOPICS IN GEOMETRY
Academic year and teacher
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- Versione italiana
- Academic year
- 2016/2017
- Teacher
- FILIPPO ALFREDO ELLIA
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/03
Training objectives
- The main goal of the course is to introduce the student to Grothendieck’s theory of schemes. Today it is almost impossible to read a research paper in algebraic geometry without knowing schemes theory. Taking as a guide the , by now classical, book of Hartshorne “Algebraic geometry”, we will propose a “quick” reading of chapters II and III of the book (chapter I being covered by the course “geometria algebrica”). We will skip the most technical results (sending back to the book) but we will focus on the material needed for a cohomological proof of the theorem of Riemann-Roch for curves (first paragraph of chapter IV). Riemann-Roch theorem, with its generalizations, is surely the fundamental result in projective algebraic geometry.
At the end of the course the student will have a precise (but incomplete!) idea of schemes theory, one of the main achievements in the mathematics of last century. The student will also be able to read research papers and start doing research in algebraic geometry. Finally he or she will have the background for a more advanced study of algebraic geometry. Prerequisites
- The following courses: Geometria 1,2; Algebra 1, algebra commutative, geometria algebrica. Some notions on categories and differential geometry (immersions, submersions, differential forms, tangent and cotangent bundles).
Course programme
- Historical introduction. Presheaves and sheaves. Affine and projective schemes. Coherent sheaves and graded modules. Cohomology. Projective curves, morphisms and linear systems. Riemann-Roch. First consequences.
Didactic methods
- This is a “reading course”. At home, students study the theory and work out the assigned exercises. During the classroom the teacher answers questions about theory and introduces the next topics. Students expose their solutions to the assigned exercises and a correction is made under the supervision of the teacher (this requires a strong effort and an active participation from the students).
Learning assessment procedures
- Oral examination. The participation to the correction of the exercises during the course will be taken into strong consideration for the final mark.
Reference texts
- Hartshorne: “Algebraic geometry” (GTM 52, Springer 1977). Notes by the teacher.
