# Pic Zero and the Poincare Bundle

For any scheme $X$, $\mbox{Pic}(X)$ is the group of line bundles on $X$.  In the case of an abelian variety $A$, there is a particularly nice subgroup $\mbox{Pic}^0(A)$.

If $L \in \mbox{Pic}^0 (A)$ then,

1. $m^*L\otimes p_1^*L^{-1} \otimes p_2^*L^{-1}$ is trivial, in particular $f,g\colon S \to A \Rightarrow (f+g)^*L \cong f^*L \otimes g^*L$
2. There is a surjection with finite kernel (an isogeney) $\phi\colon A \to \mbox{Pic}^0(A)$
3. If $M$ is a line bundle on $A\times S$ s.t. there exists $s_0 \in S$ with $M|_{A\times s_0} \in \mbox{Pic}^0(A)$ then $M|_{A\times s} \in \mbox{Pic}^0(A)$ for all $s \in S$.

### Some big results

The above claims follow pretty easily from two pretty big results:

Thm: (See-saw priciple) Let $X$ be a complete variety, and $L$ a line bundle on $X\times S$ with trivial restriction on all the fibers: $L|_{X\times s} \cong \mathcal{O}_X$.  Then $L \cong p_2^*N$ for some line bundle $N$ on $S$.

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pf(sketch): Set $L_s = L|_{X\times s}$.  The significant part of the proof is an application of cohomology with base change (see page 20-21 or J.Milne’s abel. var. notes).  The condition on the fibers of $L$ gives that $p_{2*}L$ is a locally free sheaf and in particular invertible.  Now there is always a natural map

$\alpha\colon p_2^*p_{2*}L \to L$

Set $X_s = X \times s \cong X$.  For any open set $U \subset X_s$, one can compute

$p_2^*p_{2*}L_s(U) = \mathcal{O}_X(U) \otimes_{k(s)} p_2^{-1}p_{2*}L_s(U) = \mathcal{O}_X(U) \otimes \Gamma(X, L_s)$

Using $L_s \cong \mathcal{O}_X$ and using that for a complete variety there are only constant global section (see post) I conclude $p_2^*p_{2*}L_s \to L_s$ is an isomorphism.  Its an isomorphism on fibers, and in particular an iso for any point $t \in X_s$, i.e. for every $p \in X\times S$ the map on stalks is an iso after modding out by the maximal ideal and a Nakayama’s lemma argument shows it was already an isomorphism.   So $\alpha$ is an isomorphism (since it is on stalks). QED.

Thm (of the cube): Let $X,Y$ complete and $Z$ another variety and $L$ a line bundle on $X\times Y \times Z$.  If there exists $x,y,z$ such that the restriction of $L$ to $x \times Y\times Z ,X \times y \times Z, X \times Y \times z$ is trivial, then $L$ is trivial.

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The proof is not too bad.  A rough summary: first show it holds when $X$ is a complete curve (this uses that the Jacobian of a curve is a certain moduli space and the rigidity lemma).

For the general case, let $L$ be the line bundle on $X \times Y \times Z$.  Use a complete curve $C$ to connect any $x' \in X$ to the distinguished $x \in X$ (see this post).  So have

$latex C \times Y \times Z \xrightarrow{i} X \times Y \times Z \xrightarrow{p} Z \times Z$

By the case of a curve, $i^*L$ is trivial. This shows for every $(x,z)$ the restriction of $L$ to $x \times Y \times z$ is trivial, i.e. trivial on the fibers of $p$.  By the see-saw theorem it follows $L \cong p^*N$, but $N \cong L|_{X \times y \times Z}$ which is trivial, hence $L$ is trivial.

### Corollaries

Let $p,q,r$ denote the three projections $A \times A \times A \to A$.  Then

Cor1: For any line bundle $L$ on A, the line bundle

$\begin{array}{rl} pqr(L):= & (p+q+r)^*L \\ \mbox{} & \otimes (p+q)^*L^{-1}\otimes(p+r)^*L^{-1}\otimes(q+r)^*L^{-1} \\ \mbox{} & \otimes p^*L\otimes q^*L \otimes r^*L \end{array}$

is trivial on $A \times A \times A$.

Cor2: For morphisms $f,g,h \colon S \to A$ the line bundle

$\begin{array}{rl} fgh(L):= & (f+g+h)^*L \\ \mbox{} & \otimes (f+g)^*L^{-1}\otimes(f+h)^*L^{-1}\otimes(g+h)^*L^{-1} \\ \mbox{} & \otimes f^*L\otimes g^*L \otimes h^*L \end{array}$

on $S$ is trivial.

pf: $fgh(L) = (f,g,h)^{-1}pqr(L)$, where $(f,g,h) \colon S \to A\times A \times A$.QED.

Cor3. For all $a,b \in A$ and $L$ a line bundle there is an isomorphism

$t_{a+b}^*L \otimes L \cong t_a^*L \otimes t_b^*L$

In particular, $a \mapsto t_a^*L \otimes L^{-1}$ is a homomorphism.

pf: apply the previous corollary to $id, a, b \colon A \to A$, where $a \colon A \to A$ sends everything to $a$.  The last statement follows by multiplying the above by $L^{-2}$  QED.

Given a line bundle $L$ , define $\phi_L \colon A \to \mbox{Pic}(A)$ by $\phi_L(a) = t_a^*L \otimes L^{-1}$.  The following properties are readily checked:

1. $\phi_L(a+b) = \phi_L(a)\otimes \phi_L(b) =: \phi_L(a) + \phi_L(b)$
2. $\phi_{L\otimes M}(a) = \phi_L(a)\otimes \phi_M(a) =: (\phi_L + \phi_M)(a)$
3. $\phi_{t_a^*L \otimes L^{-1}} = 0$

For the last statement it suffices to show $\phi_L = \phi_{t_b^*L}$:

$\phi_{t_b^*L}(a) = t_a^*(t_b^*L) \otimes t_b^*L^{-1}$

$= t_{a+b}^*L \otimes t_b^*L^{-1} = t_a^*L \otimes L^{-1} \otimes t_b^*L \otimes t_b^*L^{-1}$

$= t_a^*L \otimes L^{-1} = \phi_L(a)$

Finally, define $\mbox{Pic}^0(A) = \{L \in \mbox{Pic}(A) | \phi_L = 0\}$.  Set $\Lambda(L) = m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}$. Note that by 3 above, $\phi_L$ lands in $\mbox{Pic}^0$.  Now here are some more properties that are not that hard to prove

1. $L \in \mbox{Pic}^0(A)$ iff $\Lambda(L)$ is trivial (see-saw principle)
2. $f,g \colon S \to A \Rightarrow (f+g)^*L \cong f^*L \otimes g^*L$
3. $[n]^*L \cong L^n$
4. $L^n \cong \mathcal{O}_A \Rightarrow L \in \mbox{Pic}^0$  because $0 = \phi_{L^n}(a) = \phi_L(na)$ and use divisibility.
5. For $K(L) = \ker \phi_L$, if $K(L)$ is finite, then $\phi_L \colon A \to \mbox{Pic}^0(A)$ is surjective. (uses Leray Spectral sequence)

For the last bullet point in the introduction, namely If $M$ is a line bundle on $A\times S$ s.t. there exists $s_0 \in S$ with $M|_{A\times s_0} \in \mbox{Pic}^0(A)$ then $M|_{A\times s} \in \mbox{Pic}^0(A)$ for all $s \in S$.  The trick is to consider the augmented $\Lambda(L)$ on $A\times A \times S$ and use the theorem of the cube.

### Dual Variety and the Poincare Bundle

From the discussion above, for $L$ such that $K(L)$ is finite, the scheme $\hat A := A/K(L)$ can be constructed.  Its called the dual abelian variety.  There is a line bundle $\mathcal{P}$ on $A \times \hat A$ called the Poincare bundle that has the following property.  For every $\alpha \in \hat A$, $\mathcal{P}_{\alpha} \cong \alpha$, and $\mathcal{P}_{o \times \hat A}$ is trivial.  Furthermore, it is universal for this: if $L$ is a line bundle on $A \times S$ such that $L_s \in \mbox{Pic}^0(A)$ and trivial on the identity on A, then there is a morphism $f \colon S \to \hat A$ such that $L \cong (id, f)^*\mathcal{P}$.  The most popular way to construct $\mathcal{P}$ is to give decent data for $\Lambda(L)$ on $A \times A$ and the morphism $A \times A \xrightarrow{(id, \phi_L)} A \times \hat A$, see page 78 of Mumford Abelian varieties for details.

Its not hard to check in the case of $\Lambda(L)$ on $A \times A$, the morphism $A \to \hat A$ should in fact be $\phi_L \colon A \to \hat A$.  I probably should have said this earlier, but  the good $L$ to choose for $\phi_L$ are the ample line bundles, any ample one will do.