54777 - Projective Geometry

Learning outcomes

The student gets the basic elements in the theory of projective spaces. He sees affine geometry as the local aspect of projective geometry and viceversa projective geometry as a synthesis of affine phenomena.

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. Affine classification (over R and over C) and euclidean classification for conics and quadrics, with a a geometric description of their canonical forms. Plane geometry over an algebraically closed field: intersection multiplicity for a line and a curve at a point, singular points for curves and their multiplicities. Tangent line at a simple point, tangent cone. Examples of singularities: node, cusp, tacnode.  The group law on a plane smooth cubic curve in the projective plane. An outline of the classification of plane smooth projective cubics. 

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

Lectures and exercise sessions

Assessment methods

Oral examinations. During the oral examinations the student has to prove that he/she knows the main arguments treated in the course, and is able to do exercises using the instruments acquired in the course.

Teaching tools

Additional excercise sheets can be found at http://www.dm.unibo.it/~ida/annoincorso.html

Some arguments treated in this course can be found at  http://progettomatematica.dm.unibo.it/GeometriaProiettiva/hompg/hompg.htm

Links to further information

http://www.dm.unibo.it/~ida/

Office hours

See the website of Monica Idà