The Noether-Lefschetz locus of surfaces in toric threefolds

The Noether–Lefschetz theorem asserts that any curve in a very general surface X in P3 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 PΣ (with orbifold singularities) has the same Picard number as PΣ. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in PΣ 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.