B9037 - VARIETA' ALGEBRICHE

Learning outcomes

The goal of the course is to provide a robust introduction to techniques, problems and applications in algebraic geometry.

Course contents

Introduction to affine, projetive and abstract  algebraic varieties and to  their properties. Ideals, varieties, ring of functions, field of functions. Examples. Maps, Morphisms, maps  of affine and projective varieties. Abstract varieties. Examples, including weighted projective spaces.  Singular and non singular varieties. Dimensions.  Tangent space. Volume forms. The canonical divisor. Adjunction formula. The genus of a plane curve.  Rational parametrization. Birationality problems; Rationality questions. The courses  Algebraic Varitieties and Projective Geometry can be taken in the same year or in different years, in any sequence. The courses do not overlap, rather, they complement each other; the syllabi wiil be coordinated. Prerequisite: Having taken courses of the first two years of the Laurea Triennale in Matematica or equivalent.

Readings/Bibliography

M. Reid, Undergraduate Algebraic Geometry

Some research papers and other material posted on the course page on in Virtuale. In particular some  sections from the following textbooks:

I.R. Shafarevich: Basic Algebraic Geometry I, (Second Edition), Springer Verlag, 1994

D. Cox, J. Little, D. O'Shea: Ideals, varieties and algorithms.

R. Hartshorne, Algebraic Geometry 

Teaching methods

 

Presentation on Blackboard and tablet, and other digital media.

Assessment methods

 

Short (1-2 pages) essay on topics on the course, to be turned in a few days before the oral exam. The essay will be corrected, but no grade will be assigned. The oral exam will start from the written essay.

Teaching tools

Material uploaded to Virtuale.

Forum for class discussions on Virtuale.

Exercises assigned during class or posted on Virtuale.

Office hours: by appointment, via email, or agreed during class.

Office hours

See the website of Antonella Grassi