Noether-Lefschetz theory in toric varieties and its connection with Mori dream spaces

• Grant number: 23/01360-2
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: July 01, 2023
• End date: June 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Marcos Benevenuto Jardim
• Grantee: William Daniel Montoya Cataño
• Supervisor: Alex Massarenti
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Università degli Studi di Ferrara, Italy
• Associated to the scholarship: 19/23499-7 - Noether-Lefschetz theory in toric varieties, BP.PD

Abstract

Bruzzo and Grassi proved a "Noether-Lefschetz theorem for toric varieties", which claims, if Y is a 2k+1-dimensional Oda projective simplicial toric variety then for a very general quasi-smooth hypersurface X with big enough degree, one has that every (k,k)-cohomology class comes from the ambient space. The Noether-Lefschetz locus is the locus of quasi-smooth hypersurfaces with the same degree such that there exists at least a (k,k)-cohomology class that does not come from the ambient space. My Postdoctoral research has focused on the study of this geometrical object. On the other hand, Mori dream spaces were introduced by Hu and Keel as a natural extension of toric varieties, in the context of the minimal model program. In their foundational paper, they provided a canonical embedding of every Mori dream space in a projective toric variety with orbifold singularities, a fact explored by Craw and Winn to construct Mori dream spaces as moduli of quiver representations. On the other hand, Castravet and Tevelev proved that some moduli spaces of parabolic vector bundles on P1 are Mori dream spaces. The main purpose of this project is to explore the connection between Noether-Lefschetz and Mori dream spaces. Another purpose is to study the relation of moduli of quiver representations and parabolic vector bundles on P1 via Mori dream spaces. (AU)