- Docente: Alessandro Gimigliano
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Moduli: Luca Migliorini (Modulo 1) Alessandro Gimigliano (Modulo 2)
- Teaching Mode: In-person learning (entirely or partially) (Modulo 1); In-person learning (entirely or partially) (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 8010)
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. 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 sessionsfor exercises
Assessment methods
An oral exam, with a short written test before, in order to get
admitted to the oral.
Teaching tools
Some arguments treated in this course can be found at http://progettomatematica.dm.unibo.it/GeometriaProiettiva/hompg/hompg.htm .
Additional excercise sheets will be posted on the web.
Office hours
See the website of Alessandro Gimigliano
See the website of Luca Migliorini