Oberwolfach Report March 2016, Toric Geometry (Abstract)

 Research project  in collaboration with U. Bruzzo:

The Noether-Lefschetz theorem states that any curve in a very general surface X in P^3 of degree d \geq 4  is a restriction of a surface in the ambient space. We show that under a condition which plays the role of the degree, a very general surface in a simplicial toric threefold has the same Picard number as  the ambient space. We study the "degree" condition in the toric setting and relate it to the Oda conjecture of surjectivity of polytopes and to Castelnuovo-Mumford regularity.
We then  consider the Noether-Lefschetz  loci  of  quasi-smooth surfaces, that is where the Picard number is higher than the very general one in  a linear system of a Cartier ample divisor,  and study their properties.