54777 - Projective Geometry

Learning outcomes

After the course, a student should know the main elements of Projective Geometry, and should be able to view affine geometry as a local aspect of the projective environment. Familiarity with the study of elementary properties of algebraic curves in the 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. Affine covering for a projective space. Projective duality. 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.

Blow ups of the affine planes. Blow ups in affine and projective spaces. Introduction to resolution of singularities. Linear systems; examples. Veronese and Segre varieties.

The irreducible cubic with a node and the one with a cusp are rational curves. An irreducible singular plane cubic is projectively equivalent to one of the two. A smooth cubic in P^2 is projectively equivalent to a cubic of equation : y^2=x(x-1)(x-a) with a different from 0,1. The configuration of the inflection points of a smooth plane cubic. Salmon's Theorem and the classes of projective equivalence for smooth plane cubics. The group law. Smooth plane cubics as complex tori.

 

The courses Projective geometry and Curves and surfaces can be taken individually or together. The courses do not overlap, rather, they complement each other; the syllabi will be coordinated.

 

Readings/Bibliography

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

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

Material published on Virtuale

 

 

Teaching methods

Classroom lessons with exercises.

Assessment methods

The assessment is about the following aims:

-to be able to explain some part of the course, showing to be able to understand its fundamental concepts and the methods of deduction:

- to be able to solve excercises about the subject of the course.

The assessment is obtained via an oral exam.

Teaching tools

Files with exercises will be posted on Virtuale.

At

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

you can find notes on part of the course and interactive exercises.

Office hours

See the website of Monica Idà

See the website of Alessandro Gimigliano

See the website of Antonella Grassi