The Noether-Lefschetz locus of surfaces in toric threefolds

The Noether-Lefschetz theorem asserts that any curve in a very general surface X in P-3 of degree d >= 4 is a restriction of a surface in the ambient space, that is, the Picard number of X is 1. We proved previously that under some conditions, which replace the condition d >= 4, a very general surface in a simplicial toric threefold P-Sigma (with orbifold singularities) has the same Picard number as P-Sigma. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in PE in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.