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The Noether–Lefschetz locus of surfaces in toric threefolds

Area di Matematica, Scuola Internazionale Superiore di Studi, Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34149 Trieste, Italy

and

David Rittenhouse Laboratory, Department of Mathematics, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104, USA

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219199717500705Cited by:6 (Source: Crossref)

Abstract

The Noether–Lefschetz theorem asserts 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, 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 $ℙ_{Σ}$ (with orbifold singularities) has the same Picard number as $ℙ_{Σ}$ . Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in $ℙ_{Σ}$ 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.

Keywords:

AMSC: 14C22, 14J70, 14M25

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