[Paper Review] The Picard Scheme

Steven L. Kleiman|arXiv (Cornell University)|Apr 1, 2005

History and Theory of Mathematics17 references24 citations

TL;DR

This paper provides a comprehensive development of the theory of the Picard scheme as envisioned by Grothendieck, building on his Bourbaki talks and commentary. It offers a detailed exposition of the foundational framework for the Picard scheme in algebraic geometry, with a 12-page historical introduction tracing its evolution from Bernoulli to Grothendieck, making it accessible to a broader audience while advancing the technical theory for specialists.

ABSTRACT

We develop in detail most of the theory of the Picard scheme that Grothendieck sketched in two Bourbaki talks and in commentaries on them. Also, we review in brief much of the rest of the theory developed by Grothendieck and by others. But we begin with a twelve-page historical introduction, which traces the development of the ideas from Bernoulli to Grothendieck, and which may appeal to a wider audience.

Motivation & Objective

• To systematically develop the theory of the Picard scheme as sketched by Grothendieck in his Bourbaki talks and commentaries.
• To provide a rigorous and comprehensive treatment of the Picard scheme in algebraic geometry, filling in technical gaps in Grothendieck's sketches.
• To make the advanced theory of the Picard scheme accessible to a wider audience through a detailed historical introduction tracing its conceptual evolution.
• To review and integrate the broader developments in the theory of the Picard scheme by Grothendieck and other mathematicians.

Proposed method

• Systematic development of the Picard scheme using foundational techniques in algebraic geometry, including group schemes and representability.
• Use of Grothendieck's framework of functors of points to define and analyze the Picard functor.
• Application of cohomological methods, particularly sheaf cohomology, to study line bundles and their moduli.
• Incorporation of results from Grothendieck's foundational work in EGA and SGA, especially on representability of functors.
• Use of descent theory and flat base change to establish properties of the Picard scheme over general base schemes.
• Integration of historical context and conceptual evolution to frame the technical developments within the broader trajectory of mathematical thought.

Experimental results

Research questions

• RQ1 How can the Picard scheme be rigorously constructed and characterized as a moduli space for line bundles on a scheme?
• RQ2 What are the necessary and sufficient conditions for the Picard functor to be representable by a scheme?
• RQ3 How does the historical development of the concept—from early ideas on line bundles to Grothendieck’s modern framework—inform the current understanding of the Picard scheme?
• RQ4 What role do cohomological and descent-theoretic techniques play in the construction and properties of the Picard scheme?
• RQ5 How do Grothendieck’s foundational insights in the Bourbaki talks relate to and extend the broader theory developed by other mathematicians?

Key findings

• The paper establishes a complete and systematic construction of the Picard scheme for proper and flat morphisms under suitable conditions, confirming representability of the Picard functor.
• It demonstrates that the Picard scheme parametrizes line bundles on a scheme in a functorial and geometrically meaningful way, providing a moduli interpretation.
• The historical introduction reveals the conceptual lineage from classical analysis and complex geometry to modern algebraic geometry, highlighting the role of duality and cohomology.
• The paper clarifies the relationship between the Picard scheme and the Picard group, showing how the scheme structure encodes information about families of line bundles.
• It confirms that the Picard scheme is a group scheme, and under appropriate conditions, it is of finite type and smooth over the base.
• The work synthesizes and extends Grothendieck’s sketches into a coherent and self-contained theory, serving as a foundational reference for the subject.