The Noether-Lefschetz locus of surfaces in toric threefolds

The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $\mathbb P^3$ of degree $d \geq 4$ 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 $d \geq 4$, a very general surface in a simplicial toric threefold $\mathbb P_\Sigma$ (with orbifold singularities) has the same Picard number as $\mathbb P_\Sigma$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $\mathbb P_\Sigma$ 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 that is "minimal" in a suitable sense.