A Noether-Lefschetz Theorem for Spectral Varieties with Applications
Xiaoyu Su, Bin Wang
TL;DR
This paper computes the Picard group of generic spectral varieties in certain line bundle spaces and applies these results to analyze Hitchin systems over a specific algebraic surface.
Contribution
It extends the Noether-Lefschetz theorem to spectral varieties and determines the generic fibers of Hitchin systems on a quintic surface.
Findings
Calculated the Picard group of spectral varieties in specified settings.
Determined the generic fibers of Hitchin systems over a quintic surface.
Extended Noether-Lefschetz results to spectral varieties.
Abstract
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Nonlinear Waves and Solitons