Projective Algebraic Geometry
A.Y. 2019/2020
Learning objectives
The aim of the course is to give an introduction to affine and projective algebraic varieties.
Expected learning outcomes
Knowledge of some elementary properties of affine and projective algebraic varieties and ability to use them in some concrete instances.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Lesson period
First semester
Course syllabus
Affine varieties. Algebraic sets, Hilbert basis Theorem, Zariski topology, ideal of an affine algebraic set, Hilbert's Nullstellensatz, irreducible spaces and algebraic sets, regular functions, properties of the coordinate ring, regular mappings, category of affine algebraic sets, function fields and rational functions, regular points.
Quasi-projective varieties and their properties. Projective space and dual projective space, homogeneous coordinates, projective algebraic sets, the projective NullStellensatz, projective closure, projection in proective spaces from linear spaces, homogeneization of affine algebraic sets. Quasi-projective varieties. Quasi-projective varieties are locally affine. Morphisms of quasi-projective varieties: main examples of regular functions in projective space. Products of quasi-projective varieties. Regular maps of projective space. Graphs of regular functions. A morphism from a projective variety is closed in the Zariski topology. Families of quasi-projective varieties. Incidence correspondences. Lines contained in a hypersurface. The Fiber dimension Theorem. Tangent spaces. Intersection multiplicity. Tangent lines and the tangent space. Smooth points. The locus of non-singular points is open. Review on the transcendence degree of a function field. Dimension of a variety. Properties of dimension. Examples. Dimension of hyperplane sections. Equivalent formulations of dimension.
Examples of projective varieties and mappings. Veronese embedding, Grassmannians, Segre products, generalizations of the twisted cubic, hypersurfaces (in particular of fixed degree), hyperquadrics, the parameter space of conics in projective plane (the notion of parameter space in general).
Rational maps. Definition and examples; composition and categories. Dimension of indeterminacy; images and graphs of rational maps. Blowing up a point in affine space. Blow up of a smooth variety along an ideal. Resolution of a rational map via blowing up: projection from linear spaces in projective space. Birational maps. Rational, unirational and rationally connected varieties. Ruled and uniruled varieties.
Divisors. Definition of Weil divisors for smooth projective varieties over the field of complex numbers. The divisor of zeros and poles. Order of vanishing. Divisor class group. Divisor and regularity. Exceptional divisor of blow ups. Linear equivalence and linear systems. From rational maps to linear systems and backwards.
Quasi-projective varieties and their properties. Projective space and dual projective space, homogeneous coordinates, projective algebraic sets, the projective NullStellensatz, projective closure, projection in proective spaces from linear spaces, homogeneization of affine algebraic sets. Quasi-projective varieties. Quasi-projective varieties are locally affine. Morphisms of quasi-projective varieties: main examples of regular functions in projective space. Products of quasi-projective varieties. Regular maps of projective space. Graphs of regular functions. A morphism from a projective variety is closed in the Zariski topology. Families of quasi-projective varieties. Incidence correspondences. Lines contained in a hypersurface. The Fiber dimension Theorem. Tangent spaces. Intersection multiplicity. Tangent lines and the tangent space. Smooth points. The locus of non-singular points is open. Review on the transcendence degree of a function field. Dimension of a variety. Properties of dimension. Examples. Dimension of hyperplane sections. Equivalent formulations of dimension.
Examples of projective varieties and mappings. Veronese embedding, Grassmannians, Segre products, generalizations of the twisted cubic, hypersurfaces (in particular of fixed degree), hyperquadrics, the parameter space of conics in projective plane (the notion of parameter space in general).
Rational maps. Definition and examples; composition and categories. Dimension of indeterminacy; images and graphs of rational maps. Blowing up a point in affine space. Blow up of a smooth variety along an ideal. Resolution of a rational map via blowing up: projection from linear spaces in projective space. Birational maps. Rational, unirational and rationally connected varieties. Ruled and uniruled varieties.
Divisors. Definition of Weil divisors for smooth projective varieties over the field of complex numbers. The divisor of zeros and poles. Order of vanishing. Divisor class group. Divisor and regularity. Exceptional divisor of blow ups. Linear equivalence and linear systems. From rational maps to linear systems and backwards.
Prerequisites for admission
The arguments of the courses of Algebra IV and Geometria IV
Teaching methods
Lessons
Teaching Resources
◦ K. Smith, Math 631 Notes Algebraic geometry (attention to typos!).
◦ R. Shafaverich, Basic Algebraic Geometry I, Springer (in particular for rational maps and the fiber dimension theorem).
◦ J. Harris, Algebraic Geometry, Springer (for blow ups and resolution of rational maps)
◦ R. Shafaverich, Basic Algebraic Geometry I, Springer (in particular for rational maps and the fiber dimension theorem).
◦ J. Harris, Algebraic Geometry, Springer (for blow ups and resolution of rational maps)
Assessment methods and Criteria
The final examination consists of an oral exam. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The examination is passed if the oral part is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
.
- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in XXX. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take N midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding XXX in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of XXX into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if all three parts (written, oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
The examination is passed if the oral part is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
.
- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in XXX. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take N midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding XXX in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of XXX into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if all three parts (written, oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor:
Bini Gilberto
Shifts:
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Professor:
Bini Gilberto