Introduce to the study of topology and geometry defined through algebraic tools. Refine the concepts in algebra through applications to the study of algebraic varieties in affine and projective spaces
Curriculum
teacher profile teaching materials
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Introduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Mutuazione: 20410449 GE410 - GEOMETRIA ALGEBRICA 1 in Matematica LM-40 LOPEZ ANGELO, VIVIANI FILIPPO
Programme
Affine SpacesZariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Core Documentation
L. CaporasoIntroduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Type of evaluation
The exam is in a seminar fashion. It consists in explaining a topic agreed with the professor in front of the other students. teacher profile teaching materials
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Introduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Mutuazione: 20410449 GE410 - GEOMETRIA ALGEBRICA 1 in Matematica LM-40 LOPEZ ANGELO, VIVIANI FILIPPO
Programme
Affine SpacesZariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Core Documentation
L. CaporasoIntroduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Type of evaluation
The exam is in a seminar fashion. It consists in explaining a topic agreed with the professor in front of the other students. teacher profile teaching materials
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Introduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Mutuazione: 20410449 GE410 - GEOMETRIA ALGEBRICA 1 in Matematica LM-40 LOPEZ ANGELO, VIVIANI FILIPPO
Programme
Affine SpacesZariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Core Documentation
L. CaporasoIntroduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Type of evaluation
The exam is in a seminar fashion. It consists in explaining a topic agreed with the professor in front of the other students. teacher profile teaching materials
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Introduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Mutuazione: 20410449 GE410 - GEOMETRIA ALGEBRICA 1 in Matematica LM-40 LOPEZ ANGELO, VIVIANI FILIPPO
Programme
Affine SpacesZariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
Core Documentation
L. CaporasoIntroduction to algebraic geometry
Notes of the course
I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994
Type of evaluation
The exam is in a seminar fashion. It consists in explaining a topic agreed with the professor in front of the other students.