Families of toric varieties.

*(English)*Zbl 1040.14014From the introduction: The purpose of this note is to present a construction for families of complete toric varieties, which extends the standard one of a single variety. The construction is very much in the spirit of D. A. Cox [J. Algebr. Geom. 4, No. 1, 17–50 (1995; Zbl 0846.14032)], where toric varieties are described as quotients of open subsets in affine spaces: Here we simply replace the affine space by the total space of a vector bundle over an arbitrary base variety.

Such families naturally occur in the study of moduli spaces of curves on toric varieties [see the author’s preprint, M. Halic, “Higher genus curves on toric varieties”, http://arXiv.org/abs/math.AG/0101100)] or, alternatively, as the space of solutions of certain vortex-type equations on a Riemann surface [see Ch. Okonek and A. Teleman, Asian J. Math. 7, No. 2, 167–198 (2003)]. Integration on these moduli spaces do furnish invariants, the so-called Hamiltonian/gauge-theoretical Gromov-Witten invariants for the toric variety we start with. The intersection theory on these spaces is rather difficult, but one needs at least the knowledge of a presentation for the cohomology ring of the moduli space. With this motivation in mind, we construct and study families of toric varieties for their own sake.

The first section is devoted to the set up the stage: We review the construction of toric varieties as quotients of open subsets in affine spaces. This task is accomplished in the second section: We prove that is possible to “paste” complete toric varieties together, and to obtain a relative version of the standard construction of toric varieties. The objects obtained in this way are categorical quotients of Zariski open subsets in the total space of a vector bundle over an arbitrary base. They are natural generalizations of the projectivized bundle of a vector bundle. The next step is to give a presentation for the cohomology and Chow rings of families of complete and simplicial toric varieties. We conclude the article with an overview of the situation appearing in the preprints mentioned above in the context of gauged sigma models for toric varieties.

Such families naturally occur in the study of moduli spaces of curves on toric varieties [see the author’s preprint, M. Halic, “Higher genus curves on toric varieties”, http://arXiv.org/abs/math.AG/0101100)] or, alternatively, as the space of solutions of certain vortex-type equations on a Riemann surface [see Ch. Okonek and A. Teleman, Asian J. Math. 7, No. 2, 167–198 (2003)]. Integration on these moduli spaces do furnish invariants, the so-called Hamiltonian/gauge-theoretical Gromov-Witten invariants for the toric variety we start with. The intersection theory on these spaces is rather difficult, but one needs at least the knowledge of a presentation for the cohomology ring of the moduli space. With this motivation in mind, we construct and study families of toric varieties for their own sake.

The first section is devoted to the set up the stage: We review the construction of toric varieties as quotients of open subsets in affine spaces. This task is accomplished in the second section: We prove that is possible to “paste” complete toric varieties together, and to obtain a relative version of the standard construction of toric varieties. The objects obtained in this way are categorical quotients of Zariski open subsets in the total space of a vector bundle over an arbitrary base. They are natural generalizations of the projectivized bundle of a vector bundle. The next step is to give a presentation for the cohomology and Chow rings of families of complete and simplicial toric varieties. We conclude the article with an overview of the situation appearing in the preprints mentioned above in the context of gauged sigma models for toric varieties.