Introduce to the study of algebraic geometry, with particular emphasis on beams, schemes and cohomology.
Curriculum
teacher profile teaching materials
Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.
Cohomology of sheaves
Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.
Cohomology of quasi-coherent and coherent sheaves on a scheme.
Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.
Invertible sheaves and linear systems
Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).
Programme
Sheaf theory and its use in on schemesPreseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.
Cohomology of sheaves
Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.
Cohomology of quasi-coherent and coherent sheaves on a scheme.
Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.
Invertible sheaves and linear systems
Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.
Core Documentation
Notes from Prof. Lopez, Prof. SernesiR. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).
Type of delivery of the course
lessons in class, homework exercises and exercises classes. The website of reference is Moodle: https://matematicafisica.el.uniroma3.it/course/index.php?categoryid=16 Class material will be uploaded on Moodle. Lessons and exercises classes will be both in streaming and recorded and available on Teams.Type of evaluation
The evaluation will be by the exposition, in a seminar fashion, made by the student, of a subject chosen in advance and immediately following the topics of the course. Moreover a written exercise, in the same subject, will be required. teacher profile teaching materials
Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.
Cohomology of sheaves
Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.
Cohomology of quasi-coherent and coherent sheaves on a scheme.
Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.
Invertible sheaves and linear systems
Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).
Programme
Sheaf theory and its use in on schemesPreseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.
Cohomology of sheaves
Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.
Cohomology of quasi-coherent and coherent sheaves on a scheme.
Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.
Invertible sheaves and linear systems
Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.
Core Documentation
Notes from Prof. Lopez, Prof. SernesiR. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).
Type of delivery of the course
lessons in class, homework exercises and exercises classes. The website of reference is Moodle: https://matematicafisica.el.uniroma3.it/course/index.php?categoryid=16 Class material will be uploaded on Moodle. Lessons and exercises classes will be both in streaming and recorded and available on Teams.Type of evaluation
The evaluation will be by the exposition, in a seminar fashion, made by the student, of a subject chosen in advance and immediately following the topics of the course. Moreover a written exercise, in the same subject, will be required.