54777 - Projective Geometry

Course Unit Page

Academic Year 2018/2019

Learning outcomes

After the course, a student should know the main elements of Projective Geometry; should be able to view affine geometry as a local aspect of the projective enviroment. Familiarity with the study of elemetary properties of algebraic durves in real and complex projective plane.

Course contents

A historical introduction. Projective spaces: homogeneous coordinates, linearly independent points, linear subspaces. Linear subspaces in general position, skew and incident linear subspaces, equations. Projective morphisms, projectivities, the projective linear group. Points in general position, the fundamental theorem for projectivities. Cross-ratio. Affine covering for a projective space. Geometric models for P^n(R), P^1(C). Projective duality. Projective closure of an affine line. Homogeneous polynomials. Projective, affine, euclidean algebraic hypersurfaces; degree and other invariants, classification. A quick survey of quadratic forms and their classification. Projective hyperquadrics and their classification over R and over C; in particular, a geometric description of conics and quadrics in canonical form.

Readings/Bibliography

E.Sernesi: "Geometria 1", Bollati Boringhieri, Torino 1989

M.Reid: "Undergraduate Algebraic Geometry", Cambridge University Press 1988

[http://progettomatematica.dm.unibo.it/GeometriaProiettiva/hompg/hompg.htm]

Teaching methods

Classroom lessons (with exercises)

Assessment methods

A written text and then an oral exam.

Teaching tools

Files with exercises will be posted on line.

At

[http://progettomatematica.dm.unibo.it/indiceGenerale5.html]

there are notes on argument of the course and interactive exercises.

Office hours

See the website of Alessandro Gimigliano

See the website of Monica Idà