Noether-Lefschetz theory in toric varieties

• Grant number: 19/23499-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: November 01, 2020
• End date: November 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Marcos Benevenuto Jardim
• Grantee: William Daniel Montoya Cataño
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/21391-1 - Gauge theory and algebraic geometry, AP.TEM
• Associated scholarship(s): 23/01360-2 - Noether-Lefschetz theory in toric varieties and its connection with Mori dream spaces, BE.EP.PD

Abstract

In 2012 Bruzzo and Grassi proved a version of the Noether-Lefschetz Theorem for toric varieties, claiming that, on a very general quasi-smooth hypersurface X of an odd-dimensional projective simplicial toric Oda variety, with degree big enough, the (k,k)-cohomology classes of X come from the ambient space. The Noether-Lefschetz locus is the locus of quasi-smooth hypersurfaces with a fixed degree such that there exists at least one (k,k)-cohomology class which does not come from the ambient space. The candidate's PhD thesis focused on the study of this geometrical object. We state here the most important results of the candidate's thesis in order to get some context about the current research and future developments. In section 2, we state that, under suitable conditions, the dimension of every irreducible component N of the Noether--Lefschetz locus has lower and upper bounds. In section 3, continuing the study of the Noether-Lefschetz components, we claim that asymptotically, the components whose codimension is bounded from above for an effective constant consist of hypersurfaces containing a small degree k-dimensional subvariety. In section 4, we show a natural and different extension of the Noether-Lefschetz Theorem and hence of the Noether-Lefschetz locus. To get these theorems, we had to extend some classical results, machinery and ideas known for projective spaces to a more general setting, i.e., to projective simplicial toric varieties. Pushing forward those developments we expect to get some new results in different topics, mainly related with Noether-Lefschetz theory, which are presented in the last section of the research plan. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BRUZZO, UGO; MONTOYA, WILLIAM. On the Hodge conjecture for quasi-smooth intersections in toric varieties. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, JUL 2021. (19/23499-7)