# Geometry II

Academic Year 2023/2024 - Teacher: Francesco RUSSO

## Expected Learning Outcomes

The aim of the course is to allow the students to master the theories and techniques of Advanced Linear Algebra, of the geometry of affine and projective spaces, of affine and projective hypersurfaces, of  the rudiments of differentiable curves and surfaces.

The students will be able to apply these theories and techniques either to abstract or to concrete problems.

## Course Structure

The course consists of theoretical lectures by the teacher and of exercises and worked examples  by the teacher and

by the tutor during the complementary hours.

The exercise sessions contemplate a cooperative participation by the students through the exectution of simple calculations

or immediate deductions in order to verify the level of understanding of the theoretical lectures and to test how they are studying

the theoretical arguments via concrete examples, assuring both the assimiliation of the contents of the course and both

their ability in solving concrete problems also aiming to provide a full preparation for the final written and oral examinations.

Learning assessment may also be carried out on line, should the conditions require it

## Attendance of Lessons

Highly recommended.

## Detailed Course Content

The detailed program of the course is available on the web page of the course:  https://sites.google.com/view/dmiunictfrusso/geometria-ii

Brief description of the contents:

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 direction. 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, in
line with 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

SubjectsText References
11. Forme bilineari, prodotto scalare generalizzatoa)
21. Prodotto scalare reale e complesso, ortogonalità, applicazioni che conservano il prodotto scalare.a)
31. Endomorfismi autoaggiunti, matrici diagonalizzabili, teorema spettrale.a)
42. Spazi affini, sottospazi lineari, loro giacitura. Parallelismo. Intersezione e congiungente di sottospazi.a)
52. Isomorfismo di spazi affini, affinità, isometrie.a)
62. Spazi proiettivi, sottospazi lineari. Intersezione e congiungente di sottospazi.a)
72. Isomorfismo di spazi proiettivi, proiettività. Punti uniti in una proiettivitàa)
83. 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
93. 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