# Albanese dual to the Picard scheme

Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?

Edit: In fact, the following is true: $(\mathrm{Pic}^0(X)_{\mathrm{red}})^\vee = \mathrm{Alb}(X)$, and the Picard scheme is reduced (and then smooth and an Abelian scheme) iff equality holds in $\dim H^1(X,\mathcal{O}_X) \geq \dim \mathrm{Pic}^0(X)$ ("defect of smoothness").

-

The nicest modern reference for the theory of the Albanese that I know of is the appendix to this article of S. Mochizuki.

-
Thank you. I will have a look at it. – Timo Keller Apr 19 '13 at 12:56

I think there are many places where this is written in detail.

A good reference is Birkenake-Lange's book Complex Abelian Varieties, Proposition 11.11.6 (page 357), where the statement $\textrm{Pic}^0(M)=\widehat{\textrm{Alb}(M)}$ is proven for any smooth projective variety $M$.

-
Thanks. Unfortunately, I am interested in the result in positive characteristic. – Timo Keller Apr 19 '13 at 12:55

The original reference is: On Picard Varieties, Wei-Liang Chow, American Journal of Mathematics 74, 895-909 (1952), with references to earlier work by Weil and Igusa.

-
Thank you. I will have a look at it. – Timo Keller Apr 19 '13 at 12:55