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Noether-Lefschetz theory in toric varieties

Grant number: 19/23499-7
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): November 01, 2020
Effective date (End): October 31, 2022
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal researcher:Marcos Benevenuto Jardim
Grantee:William Daniel Montoya Cataño
Home 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

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)