In this lecture I will speak about the fundamental curve "in family". Here the family is parametrized by a characteristic p perfectoid space.

I will explain the structure of the Picard stack of line bundles in this context. Moreover I will explain the structure of positive degree effective divisors on the curve as a diamond in the sens of Scholze.

Bio: Laurent Fargues is Directeur de Recherche at CNRS/ Institut de mathe ́matiques de Jussieu. He is a leading expert in questions on the local Langlands correspondence, Rapoport-Zink spaces, and p-adic Hodge theory, and has made striking contributions to all of these areas. His work is characterized by a unique vision of how these subjects are related, and his insights are shaping these fields. He received the "Petit d’Ormoy, Carrière, Thébault", prize given by the French Academy of Science. He is an invited sessional speaker at the upcoming 2018 ICM at Rio de Janeiro, Brazil.