Topic of the course
The course will focus on algebraic surfaces. The first part will concern smooth
ones, mostly following ”Complex Algebraic Surfaces” by Arnaud Beauville, while
the second part will focus on singular surfaces and their resolutions.
Formative objectives
In the first part of the course (8 lectures) the students will learn the basics of
the theory of algebraic surfaces. The main topics are
• Riemann-Roch theorem.
• Rational maps and Castelnuovo’s theorem.
• Surfaces classification.
• Examples with a special focus on rational and ruled surfaces.
The second part of the course (2 lectures) will concern singular surfaces
of ADE type. The students will learn their main properties and the main
techniques for solving their singularities.
Prerequisites
Basic algebraic geometry ( definitions and main properties of varieties, schemes
and sheaves).
Plan of the course
Lecture 1 Introduction and first examples.
Lecture 2 Riemann-Roch theorem and applications.
Lecture 3 Riemann-Roch theorem and applications.
Lecture 4 Rational maps.
Lecture 5 Rational maps.
Lecture 6 Castelnuovo’s contraction theorem.
Lecture 7 Ruled surfaces.
Lecture 8 Rational surfaces.
Lecture 9 ADE singularities.
Lecture 10 Blow-up of fat points.
Exam
The exam will consist in the presentation of a topic previously agreed with the
lecturer