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Explore advanced research in algebraic geometry through this mathematical lecture examining the Picard ranks of non-ordinary K3 surfaces over finite fields. Delve into joint work with Maulik-Tang demonstrating that for non-isotrivial families of K3 surfaces where the generic surface is ordinary, the locus of Picard rank jumps forms an infinite set under compactness assumptions. Investigate the contrasting behavior of non-ordinary families, where examples exist with both empty and finite jump loci. Learn about ongoing collaborative research with Ruofan Jiang and Ziquan Yang that provides a complete classification of when the Picard rank jump locus is infinite, specifically under the hypothesis that the endomorphism ring of the generic Kuga-Satake abelian variety is determined by the generic Picard rank. Gain insights into the intersection of arithmetic geometry, K3 surface theory, and the behavior of Picard ranks in families of algebraic surfaces defined over finite fields.
