Geometry II
Academic Year 2024/2025 - Teacher:
Francesco RUSSO
Expected Learning Outcomes
Educational Aims:
The aim of the course is to complete the knowledge of linear algebra of the Geometry 1 course by presenting some fundamental concepts of this theory, as well as providing an almost complete treatment of the geometry of affine and Euclidean spaces on the one hand and that of projective spaces on the other hand in the spirit of the Erlangen Program developed by F. Klein (study of equivalence classes through a given group of transformations that act on the ambient space). The tools introduced are also used to study the geometry of algebraic hypersurfaces in various settings with particular emphasis on the case of quadratic entities in arbitrary dimension and on the local and global properties of plane algebraic curves.
Expected Learning Skills:
At the end of the course the student will have to demonstrate knowledge of the new tools of advanced linear algebra and geometry. Furthermore, the student must be able to adequately understand and solve linear algebra and geometric problems considered in the various settings, both from a theoretical point of view (development of a rigorous mathematical language; assimilation of definitions, theorems and the main ideas of the proofs) and from a practical point of view (solving exercises in written examination).
Course Structure
The teaching consists of theoretical lectures held by the teacher, practical exercises and the detailed study of examples, the latter explored in depth by the teacher or by tutor teachers in addition to the course.
The practical exercises require cooperative participation by the students through the performance of simple calculations and immediate deductions to verify the degree of understanding of the topics covered and how much they are actually studying the theoretical topics learned. These practical activities ensure equal assimilation of the course contents and the ability to solve concrete problems also in view of passing the written test and the oral test.
Required Prerequisites
The student should at least know the contents of the Geometry 1 course and it is strongly recommended to have learned the basic concepts of the Algebra course (group, ring, field).
Attendance of Lessons
Highly recommended.
Detailed Course Content
Bilinear forms, generalized inner pooducts. Real and complex inner products, ortogonality, linear maps preserving inner product.
Adjoint endomorphisms, normal matrices, Spectral Theorem for normal operators.
Affine spaces, linear subspaces and their directions. Parallelism. Intersection and linear span of subspaces. Dimension and codimension of subspaces.
Isomorphisms of affine spaces, isometries. Projective spaces, linear subspaces. Intersection and join of linear spaces. Dimension and codimension of linear spaces.
Isomorphisms of projective spaces, projective transformations, fixed points of a projective transformation.
Affine and projective hypersurfaces and their relations. Intersection with a line, simple and multiple points. Tangent lines to a hypersurface in a point,
tangent cone, tangent space and their equations. Bezout Theorem and its applications. Flexes of a curve and the Hessian curve. Polarity and its geometrical meaning.
Group structure on a plane cubic curve and some applications.
Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, according to the programme planned and outlined in the syllabus
Textbook Information
a) E. Sernesi: Geometria I, Bollati Boringhieri, Torino
b) E. Sernesi: Geometria II, Bollati Boringhieri, Torino.
c) C. Ciliberto: Algebra Lineare, Bollati Boringhieri, Torino
Latex written notes of the course are freely available at the internet page of the course : https://drive.google.com/file/d/1pCkZRyqSCyawPybrHTLgh3gN8IQD4hca/view
Moreover, the previous web page contains many exercise and written exams, most of them also fully solved.
Course Planning
| | Subjects | Text References |
|---|
| 1 | 1. Forme bilineari, prodotto scalare generalizzato | a) |
| 2 | 1. Prodotto scalare reale e complesso, ortogonalità, applicazioni che conservano il prodotto scalare. | a) |
| 3 | 1. Endomorfismi autoaggiunti, matrici diagonalizzabili, teorema spettrale. | a) |
| 4 | 2. Spazi affini, sottospazi lineari, loro giacitura. Parallelismo. Intersezione e congiungente di sottospazi. | a) |
| 5 | 2. Isomorfismo di spazi affini, affinità, isometrie. | a) |
| 6 | 2. Spazi proiettivi, sottospazi lineari. Intersezione e congiungente di sottospazi. | a) |
| 7 | 2. Isomorfismo di spazi proiettivi, proiettività. Punti uniti in una proiettività | a) |
| 8 | 3. Ipersuperficie affini e proiettive, connessioni. Intersezione con una retta, punti semplici e punti multipli. Rette tangenti in un punto, cono tangente, spazio tangente e loro equazioni. | Note di corso |
| 9 | 3. Teorema di Bezout e applicazioni. Flessi e curva hessiana. Polarità e suo significato geometrico. Struttura di gruppo sui punti di una cubica piana, applicazioni geometriche. | Note di corso |
Learning Assessment
Learning Assessment Procedures
The exam consists of a three-hour written test and an oral exam of variable length (also depending on the outcome of the written test).
Rigorous preparation for the written test will allow the student to apply the theoretical tools learned in concrete examples.
The oral exam will require a clear and precise exposition of the theoretical content developed during the course, verifying the maturation of the students' learning, their expository ability and the degree of elaboration of the content achieved.
Students who take a written test with a grade lower than 12/30 are not admitted to take the oral exam and will have to repeat the written exam.
The first in-progress test will be held during the break between semesters. It consists of a three-hour written test and an oral exam on the course content relating to the first semester and can therefore be considered equivalent to 6CFU.
The second in-progress test will be held at the end of the course, usually in conjunction with the first available exam session in June. It consists of a three-hour written test and an oral test, which will focus on the contents of the course relating to the second semester and can therefore be considered equivalent to 6CFU.
The learning assessment may also be carried out electronically, if conditions require it. In this case, the duration of the written test may be subject to change.
Generally, grades will be assigned according to the following scheme:
- not approved: the student has not acquired the basic concepts and is unable to complete the exercises.
- 18-23: the student demonstrates minimal mastery of the basic concepts, his/her skills in expounding and connecting the contents are modest, he/she can solve simple exercises
- 24-27: the student demonstrates good mastery of the contents of the course, his/her skills in expounding and connecting the contents are good, he/she solves the exercises with few errors
- 28-30 cum laude: the student has acquired all the contents of the course and is able to fully explain them and connect them with a critical spirit; solves the exercises completely and without errors.
Information for students with disabilities and/or DSA:
To ensure equal opportunities and in compliance with current laws, interested students can request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.
It is also possible to contact the CInAP (Center for Active and Participated Integration - Services for Disabilities and/or DSA) contact teacher of the Department or the President of the Course of Studies.
Examples of frequently asked questions and / or exercises
On the Microsoft Team of the Course it is possible to see and download the written exans assigned in previous years.
The questions in the oral interview aim to ascertain the actual understanding of the statements of the main theorems and their applications rather than a straightforward verification of a notional knowledge of the proofs. They will be formulated on some of the topics of the Program maintaining a balance between the contents of Advanced Linear Algebra and those of Geometry.
