formal Picard group in nLab

Contents

Contents

Idea

Given an algebraic variety with Picard scheme Pic XPic_X, if the connected component Pic X 0Pic_X^0 is a smooth scheme then the completion of Pic XPic_X at its neutral global point is a formal group. This is called the formal Picard group of XX. (ArtinMazur 77, Liedtke 14, example 6.13)

This construction is the special case of the general construction of Artin-Mazur formal groups for n=1n = 1 (see also this Remark at elliptic spectrum). The next case is called the formal Brauer group.

• Picard stack

References

The original account of the construction of formal Picard groups is

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 numdam, MR56:15663

Modern reviews include

• Christian Liedtke, example 6.13 in Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem (arXiv.1403.2538)