Let be a projective variety over the algebraically closed field , endowed with a basepoint . In the previous post, we saw how to define the **Picard scheme** of : a map from a -scheme into is the same thing as a line bundle on together with a trivialization on . Equivalently, is the sheafification (in the Zariski topology, even) of the functor

so we could have defined the functor without a basepoint.

We’d like to understand the local structure of (or, equivalently, of ), and, as with moduli schemes in general, deformation theory is a basic tool. For example, we’d like to understand the tangent space to at the origin . The tangent space (this works for any scheme) can be identified with

In other words, we have to look at maps which restrict to zero on the closed point. By the modular interpretation of , we find that

in other words, we have to understand the possible deformations of the trivial line bundle over to the dual numbers. We can do that using the exponential sequence

where the first map sends . Applying the long exact sequence in cohomology (notice that the last map has a section), we conclude:

Theorem 3The tangent space to is identified with .

In particular, we should expect the dimension of to be exactly the Hodge number , as we found earlier over using transcendental methods. Unfortunately, this is not true in general. The dimension of the tangent space of a scheme takes into account only the infinitesimal structure, but does not see the actual dimension (unless the scheme is smooth).

However, in *characteristic zero*, a theorem of Cartier (see for instance this post) states that group schemes are smooth, and in this case we can conclude that . In characteristic , this is not true in general. Later in this post, I’ll describe a counterexample of Igusa.

**3. Smoothness in dimension one**

As a test case, let’s check that the Picard scheme of a one-dimensional scheme is always smooth: for a smooth curve, it is the classical Jacobian variety. In fact, smoothness can be checked by the infinitesimal lifting criterion. It suffices to show that whenever is a local artinian ring, and is a square-zero extension by some square-zero ideal , then we can always solve a lifting problem

Equivalently, we need to show that the map

is a surjection: we can lift families of line bundles along square-zero extensions.

To do this, we consider the short exact sequence

where the first map is . Applying and looking at the long exact sequence, we find that since on the one-dimensional scheme , the map

is surjective. But this is exactly what we wanted to prove. We conclude:

Theorem 4For a one-dimensional, reduced, proper scheme over , is a smooth scheme whose components are quasi-projective.

When itself is smooth, then an argument via the valuative criterion shows that is projective, hence an abelian variety: the Jacobian variety of .

**4. The Albanese variety**

Let be a pointed, smooth variety. As before, there is a scheme such that maps are in natural bijection with line bundles over trivialized over . We can consider the connected component at the identity.

By the universal property, there is a line bundle over with a trivialization on . Now, need not itself be a variety, but it is a quasi-projective -group scheme.

Proposition 5If is smooth, then is projective.

*Proof:* One way to prove this is to use the valuative criterion to argue that if is a discrete valuation ring (over ) with quotient field , then by an argument using divisors; see this MO discussion.

In particular, we can form the reduction , and that will be an abelian variety. We thus get a line bundle on which is trivialized on . Moreover, by definition, we can trivialize the line bundle on , where is image of the identity in .

It follows that our situation is completely symmetric. That is, rather than thinking of as being a family of line bundles on parametrized by , we can think of it as a family of line bundles on parametrized by . The conclusion is that we get a map

which sends to zero. This map is natural in .

Although we have not proved it yet, for an *abelian variety*, this map is an isomorphism: the Picard scheme is in that case always smooth, and the map above is the biduality isomorphism.

Definition 6We define to be theAlbanese varietyof (in fact, the second is redundant).

In particular, we have a natural map

of pointed varieties (where an abelian variety is pointed at zero), which is an isomorphism if itself is an abelian variety. It follows that the Albanese variety is the universal abelian variety generated by : that is, it is the left adjoint of the forgetful functor from abelian varieties to pointed varieties.

**5. Igusa surfaces**

Finally, let’s describe an example, due to Igusa, where the Picard scheme is nonreduced (and so has dimension smaller than expected). I learned this example from these notes of Jesse Kass.

Let be an *ordinary* elliptic curve over an algebraically field of characteristic two. In this case, there is a unique nonzero 2-torsion point , and we can define the involution

on . The involution acts without fixed points, so we can form the quotient and get a smooth projective surface .

The basic observation is that it is very difficult for to map to an abelian variety (let’s point with the image of ). In fact, a map

for an abelian variety, is the same as a pointed map such that

Since is a pointed map of abelian varieties, it is a homomorphism. We conclude that , so that entirely kills the second factor and the image of (which is the same as the image of ) has dimension one.

In fact, it’s even better: we find that , so factors through the quotient (which is a -quotient of , hence an elliptic curve). Since there is a pointed map

we conclude that must be the universal abelian variety that maps into: that is,

However, has the same dimension as : any abelian variety is *isogeneous *to its dual. It follows that

Conversely, the expected dimension from deformation theory is at least two. We have a spectral sequence

which implies that

since acts trivially in cohomology.

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