The Noether–Lefschetz locus of surfaces in toric threefolds

Abstract

The Noether–Lefschetz theorem asserts that any curve in a very general surface [Formula: see text] in [Formula: see text] of degree [Formula: see text] is a restriction of a surface in the ambient space, that is, the Picard number of [Formula: see text] is [Formula: see text]. We proved previously that under some conditions, which replace the condition [Formula: see text], a very general surface in a simplicial toric threefold [Formula: see text] (with orbifold singularities) has the same Picard number as [Formula: see text]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [Formula: see text] in a linear system of a Cartier ample divisor with respect to a [Formula: see text]-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.