Module page
Istituzioni di Geometria Superiore
| academic year: | 2013/2014 |
| instructor: | Enrico Arbarello |
| degree course: | Mathematics (magistrale), I year |
| type of training activity: | caratterizzante |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/03 Geometria |
| teaching language: | italiano |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
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- Complex varieties.
- Riemann surfaces and algebraic curves.
- Resolution of singularities of plane algebraic curves.
- Basic properties of singular homology.
- Ramified covers and Hurwitz theorem.
- De Rahm's theorem.
- The Laplace operator on Riemann surfaces.
- The Hodge theorem for Riemann surfaces.
- The Uniformization theorem.
- The Riemann-Roch theorem for compact Riemann surfaces and applications.
- The Picard group and the Jacobian of a compact Riemann surface.
- Abel's and Jacobi's theorems.
- Complex varieties Riemann surfaces and algebraic curves. Resolution of singularities of plane agebraic curves. Basic properties of singular homology. Ramified covers and Hurwitz theorem. De Rham's theorem. Laplace operator on Riemann surfaces. The Hodge theorem. Uniformization theorem. The Riemann-Roch theorem and applications.
Type of course: standard
Knowledge and understanding: Successful students will be familiar with the basic theory of Riemann surfaces and algebraic curves.
Skills and attributes: Succesfull students will be able to solve elementary problems on Riemann surfaces and algebraic curves, such as: computing the genus of a Riemann surface, solving singularities of plane curves, finding holomorphic differentials, determining the automorphism group of a Riemann surface, mapping compact Riemann surfaces in projective spaces, using the Hurwitz theorem, the Riemann-Roch theorem and Abel's theorem.
Personal study: the percentage of personal study required by this course is the 65% of the total.