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Academic Year 2023/2024 - Teacher: GIOVANNI STAGLIANO'

Expected Learning Outcomes

The course aims to give an introduction to the basic theories and techniques of modern Algebraic Geometry, deepening the theory of affine and projective algebraic varieties, their morphisms and isomorphisms with applications to the study of their intrinsic and extrinsic properties (related to one of their set dive).

The course also aims to refine the ability to abstract and, on the other hand, show how a good theoretical knowledge allows you to develop significant application tools.

In particular, the course aims to enable students to acquire the following skills:

1) Knowledge of results and fundamental methods of the theory of algebraic varieties. Ability to read, understand and deepen a topic of mathematical literature and present it clearly and accurately. Ability to understand problems and extract their substantial elements.

2) Ability to construct or solve examples or exercises and to tackle new theoretical problems, researching the most suitable techniques and applying them appropriately.

3) Be able to produce proposals aimed at correctly interpreting complex problems in the field of Algebraic Geometry and its applications. Be able to autonomously formulate pertinent judgments on the applicability of algebro-geometry models to theoretical and/or concrete situations.

4) Ability to present arguments, problems, ideas and solutions, both one's own and those of others, in mathematical terms and their conclusions, with clarity and accuracy and in a manner appropriate to the listeners to whom one addresses, both orally and in writing . Ability to clearly justify the choice of strategies, methods and contents, as well as the computational tools adopted.

5) Read and deepen a topic of the Algebraic Geometry literature. Independently deal with the systematic study of topics related to algebraic varieties not previously explored.

Course Structure

Besides the theoretical lectures developing the foundations of the subject there will be numerous exercise sessions,  where the students will solve at the blackboard exercises from preassigned lists and work out the properties of relevant explicit examples  in order to have a solid basis of objects on which to test the abstract theories.

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in
force, interested students can ask for a personal interview in order to
plan any compensatory and / or compensatory measures, based on the
didactic objectives and specific needs. It is also possible to contact
the referent teacher CInAP (Center for Active and Participated
Integration - Services for Disabilities and / or SLD) of our Department,
prof. Filippo Stanco.

Required Prerequisites

Strongly recommended: Institutions of Higher Algebra.

Attendance of Lessons

Strongly recommended.

Detailed Course Content

I) -- Affine and projective algebraic sets. Zariski topology on affine and projective spaces. Correspondence between affine algebraic sets and radical ideal in a polynomial ring (algebraically closed field). Irreducible algebraic sets and correspondence with prime ideals. Coordinate ring of an affine algebraic variety and of a projective variety. Decomposition of an algebraic set into irreducible components and its relations with primary decomposition of an ideal. Dimension of an algebraic variety: topological and algebraic definition.

II) -- Regular functions on a quasi-projective variety: definition and first properties. Examples and applications. Morphisms between varieties: definition and first properties. Examples and applications. Local ring of regular functions on a variety: definition and first properties. Rational functions on a variety: definition and first properties. Rational (and birational) maps between algebraic varieties: definitions and first examples. Correspondence between dominant rational maps and homomorphisms of their function fields. Regular functions on a projective varieties and applications.

III) -- Product of algebraic varieties: universal property, existence and unicity. Examples and applications: graph morphism, diagonal morphism, decomposition of a morphism via
its graph morphism and projections. Fundamental Theorem of Elimination Theory. Examples and applications.

IV) -- Non-singular point on an algebraic variety: extrinsic and intrinsic definition. Singular locus. Blow-up of a variety at a point. Tangent cone and tangent space to a variety at a point: extrinsic and intrinsic definition. Examples and applications. Definition of multiplicity of a point on a variety. Comparison between the tangent cone and the tangent space at a point: non-singularity criterion. 

V) -- Theorem on the dimension of the fibers of a morphism. Applications. Irreducibility Criterion. Applications to the study of lines on superfaces in projective space with special regard to the case of cubics. Dual variety and Bertini Theorem.

Textbook Information

1) R. Hartshorne. Algebraic Geometry. Springer Verlag, 1977.

2) I. R. Shafarevich. Basic Algebraic Geometry. Springer-Verlag, 2013.

3) D. Mumford. The Red Book of Varieties and Schemes. Springer Verlag, 1988.

4) D. A. Cox, J. Little, D. O’shea. Using Algebraic Geometry. Springer Verlag, 2005.

Course Planning

 SubjectsText References
1Insiemi algebrici affini e proiettivi. Topologia di Zariski sullo spazio affine e proiettivo. Corrispondenza tra insieme algebrici affini e ideali radicali di un anello di polinomi (campo algebricamente chiuso).1), 2) e 3)
2Insiemi algebrici irriducibili e loro corrispondenza con gli ideali primi di un anello di polinomi. Anello delle coordinate di una varietà affine e di una varietà proiettiva.1), 2) e 3)
3Decomposizione di un insieme algebrico in componenti irriducibili e legami con la decomposizione primaria di un ideale. Dimensione di una varietà algebrica: versione topologica e algebrica.1), 2) e 3)
4 Funzioni regolari su una varietà algebrica quasi-proiettiva: definizione e prime proprietà. Esempi e applicazioni. Morfismi tra varietà: definizione e prime proprietà. Esempi e applicazioni.1), 2) e 3)
5Anello locale delle funzioni regolari su una varietà: definizioni e prime proprietà. Funzioni razionali su una varietà: definizione e prime proprietà. Applicazioni razionali (e birazionali) tra varietà: definizioni e prime proprietà.1), 2) e 3)
6Corrispondenza tra applicazioni razionali dominanti e omomorfismi dei rispettivi campi di funzioni razionali. Funzioni regolari su una varietà proiettiva e applicazioni.1), 2) e 3)
7Prodotto di varietà algebriche: proprietà universale, esistenza e unicità. Morfismo grafico, morfismo diagonale, decomposizione di un morfismo tramite morfismo grafico e proiezioni dal prodotto. Teorema Fondamentale della Teoria dell' Eliminazione.1), 2) e 3)
8 Punto non singolare di una varietà: definizione estrinseca e intrinseca. Luogo singolare. Scoppiamento di una varietà in un punto. Cono tangente e spazio tangente a una varietà in un punto: definizioni intrinseche e estrinseche. 1), 2) e 3)
9Definizione di molteplicità algebrica di un punto su una varietà. Confronto tra cono tangente e spazio tangente: criterio di non-singolarità. 1), 2) e 3)
10Teorema della Dimensione delle Fibre. Applicazioni. Criterio Irriducibilità. Applicazione allo studio delle rette su superficie in $\mathbb P^3$ con particolare riguardo al caso cubico.Materiale didattico
11Varietà duale e Teorema di Bertini. Varietà di spazi plurisecanti: definizioni e esempi. Mappa di Gauss: definizione e esempi.Materiale didattico

Learning Assessment

Learning Assessment Procedures

Oral exam.

During the course exercises are assigned, solved in class by the students and which contribute to the final passing mark of the course.