54777 - Projective Geometry

Learning outcomes

The student gets the basicelements 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 forP^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. Affineclassification (over R and over C) and euclidean classification for conics and quadrics, with aa 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 ofplane 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 examination

Teaching tools

Additional excercise sheets can be found athttp://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

See the website of Luca Migliorini

See the website of Mirella Manaresi