Room P3.10, Mathematics Building

Ciro Ciliberto, Università di Roma II

Degeneration techniques in algebraic geometry

In the first three lectures I will thouch these themes:

  1. generalities on divisors and linear systems on a surface;
  2. (birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);
  3. generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.
  4. I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.