# What is known about the Picard scheme of a complete toric variety over C?

Let $X$ be a complete toric variety over a field $k$. Its Picard scheme is defined to be the scheme representing the functor $Pic_{(X/k)(\text{fppf})}$, where $Pic_{X/k}: Sch_k^{op}\to Set$ sends a scheme $T$ over $k$ to the scheme $Pic(X\times T)/p_2^*Pic(T)$, and "(fppf)" denotes the sheafification of this functor w.r.t the fppf topology.

It is known that in this case the Picard scheme exists.

I want to know if there is any good combinatorial description of the Picard scheme. (We may assume $k=\mathbb{C}$ if this helps.)

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I suspect that the Picard scheme is the constant group scheme given by the combinatorial description of the Picard group. –  ACL Apr 22 '11 at 6:58
$H^1(X,\mathbb{O})=0$ because $X$ is rational (it contains a torus). So $Pic^0(X)$ is trivial. –  Donu Arapura Apr 22 '11 at 13:32
The font came out wrong, that's $\mathcal{O}$. –  Donu Arapura Apr 22 '11 at 13:33
@Donu But isn't $CP^1$ with 0 and $\infty$ identified a toric variety? This has $Pic^0 = C^*$. –  Jim Bryan Apr 22 '11 at 14:50
Jim, good point. I should say for $X$ normal complete toric, although I had the impression that normality is sometimes included in the definition of toric variety. –  Donu Arapura Apr 22 '11 at 15:38