Ugo Bruzzo:
The Noether-Lefschetz problem for normal threefolds

Abstract: The classical Noether-Lefschetz problem is about the Picard number of surfaces in P3.
For d \ge 4, if S(d) is the locus of smooth surfaces in the linear system O(d) on P3, the very general surface in S(d) has Picard number one. The Noether-Lefschetz locus NL(d) is the locus in S(d) whose points represent surfaces with Picard number greater than one. It is possible to estimate the codimensions of the components of NL(d), and show that their codimension is related to geometric properties of the surfaces they describe.

Over the last few years, in collaboration with A. Grassi, and more recently with A.F. Lopez, I have investigated generalizations to surfaces in normal, Q-factorial, possibly toric 3-folds. My talk will be devoted to describe some of the results we obtained.