8. Hilbert Schemes of Points on Curves: We survey a few of the known properties of Hilbert schemes of points on curves.

If X is a projective scheme over C, let X^{[d]} denote the Hilbert scheme of subschemes of X of constant Hilbert polynomial d (i.e. of length d).

When C is a compact Riemann surface then C^{[d]} is the symmetric product, a nonsingular complex projective variety of dimension d.

Theorem (Macdonald): The Betti numbers b_{i}(C^{[d]}) = dim H^{i}(C^{[d]}, Q) are computed by the following generating function:

∑_{d≥0} ∑_{i≥0} b_{i}(C^{[d]})t^{i}q^{d} = \frac{(1 + tq)^{2g}}{(1 − q)(1 − t^{2}q)}

Given a coherent sheaf on C^{[d]}, there is a natural cycle map:

A^{∗}(C^{[d]}) → H^{2∗}(C^{[d]}, C)

i.e. Chern classes on C^{[d]} determine cohomology classes. We may use the universal family to construct several of these.

Recall that the universal family is a subscheme:

D ⊂ C^{[d]} × C

which is flat over C^{[d]} with fibers equal to the subschemes of C of length d. Let p and q be the projection maps to C^{[d]} and C respectively.

It follows from cohomology and base change that p_{∗}q^{∗}O_{C} is locally free on C^{[d]} of rank d. The half-diagonal class \frac{1}{2}δ is defined by:

\frac{1}{2}δ := −c_{1}(p_{∗}q^{∗}O_{C})

One can show that δ ∈ A^{1}(C^{[d]}) is represented by the “big diagonal”...the closed subvariety parametrizing schemes that contain a point of multiplicity two or more. This Chern class is present on any Hilbert scheme of points.

Choose a closed point p_{0} ∈ C. Then there are induced closed immersions:

p_{0} ∈ C ↪ C^{[2]} ↪ C^{[3]} ↪ ...

given by “addition of the point p_{0}.” More accurately, when we take the quotient of the inclusion of products:

C^{d−1} × {p_{0}} ⊂ C^{d}

by the action of the symmetric group, we get the desired closed immersion of symmetric products. We define:

x_{p_{0}} := [C^{[d−1]}] ∈ A^{1}(C^{[d]})

and note that

[C^{[d]}] ∩ x_{p_{0}}^{e} = [C^{[d−e]}]

The classes [C^{[d−1]}] ∈ A^{1}(C^{[d]}) depend upon p_{0} (when C has positive genus) but their images in H^{2}(C^{[d]}, C) do not. We will let x denote the image.

Finally, we may use the fact that D ⊂ C^{[d]}×C is a Cartier divisor to define the determinantal class on C^{[d]}. We start first with the case d = g − 1. We define:

θ := −c_{1}(p_{!}O_{C^{[g−1]}×C}(D)) = c_{1}(R^{1}p_{∗}O(D)) − c_{1}(p_{∗}O(D))

If we choose a “base divisor” D_{0} = p_{1} +...+p_{g} consisting of g distinct points, then the sequence:

0 → O(D) → O(D + q^{∗}D_{0}) → O_{q^{−1}D_{0}}(D + q^{∗}D_{0}) → 0

pushes forward to a resolution:

0 → p_{∗}O(D) → E → F → R^{1}p_{∗}O(D) → 0

where E and F are vector bundles of rank g, and then:

θ = c_{1}(F) − c_{1}(E)

We get well-defined cohomology classes θ ∈ H^{2}(C^{[d]}, C) for all d by letting:

θ := c_{1}(p_{!}O(D + q^{∗}D_{0}))

for any divisor D_{0} of degree g − 1 − d.

These three divisor classes on C^{[d]} have very different properties.

(i) Each x_{p_{0}} ∈ A^{1}(C^{[d]}) is ample.

Proof: x_{p_{0}} pulls back to the ample divisor ∑ π_{i}^{∗}p_{0} on the product C^{d} (where π_{i} : C^{d} → C is the projection). The cohomological criterion for ampleness then implies x_{p_{0}} is ample, because C^{d} → C^{[d]} is a finite map.

(ii) H^{0}(C^{[d]}, O_{C^{[d]}}(\frac{n}{2}δ)) = 0 for odd n and is one-dimensional for even n.

(iii) θ is pulled back from an ample divisor on the Picard group. More precisely, each Pic^{d}(C) is an abelian variety of dimension g parametrizing the isomorphism classes of line bundles on C of degree d. There are Abel-Jacobi maps C^{[d]} → Pic^{d}(C) for each d sending D to O_{C}(D). The fibers of the Abel-Jacobi maps are the linear series |D|. The Abel-Jacobi maps are surjective when d ≥ g by Riemann-Roch. Tensoring by O_{C}(p_{0}) gives isomorphisms Pic^{d}(C) →^{∼} Pic^{d+1}(C) and a commuting diagram:

C  ↪ C^{[2]} ↪ ⋯ ↪ C^{[g]} ↪ ⋯ ↪ C^{[2g−1]}
↓ ↓ ↓ ↓
Pic^{1}(C) →^{∼} Pic^{2}(C) →^{∼} ⋯ →^{∼} Pic^{g}(C) →^{∼} ⋯ →^{∼} Pic^{2g−1}(C)

(degree 2g − 1 is significant because at that point and thereafter, the Abel-Jacobi map becomes a projective bundle)

On Pic^{d}(C) × C there is a universal line bundle L with the property that when restricted to a fiber of the projection map p to Pic^{d}(C), L is isomorphic to the line bundle parametrized by the point of Pic^{d}(C). On Pic^{g−1}(C), we can define a θ ∈ A^{1} determinantally as earlier. Namely, we let:

θ = −c_{1}(p_{!}L)

In this case, however, p_{∗}L = 0 and R^{1}p_{∗}L is supported (and generically of rank one along) the image of C^{[g−1]}. It follows that on Pic^{g−1}(C), θ is the push-forward of [C^{[g−1]}] under the Abel-Jacobi map, and on C^{[g−1]}, θ is the first Chern class of the the normal sheaf to the Abel-Jacobi map. One can prove that θ (defined as before but on Pic^{d}(C)) is ample on each Pic^{d}(C). Thus although θ is not ample on C^{[d]} whenever the Abel-Jacobi map has positive-dimensional fibers, it is the case that a high multiple of θ is basepoint free with an explicit image in projective space.

Theorem: The image of x and θ in H^{2}(C^{[d]}, Q) generate all the Chern classes of C^{[d]}, if C is “of general modulus.”

Example:

\frac{1}{2}δ = (d + g − 1)x − θ

One can similarly obtain formulas for the “smaller” diagonals ([ACGH]).

Another Example: If C is of genus g and general modulus, d ≤ 2g − 2, and r is a non-negative integer with the property that (r + 1)(g − d + r) ≤ g then in particular, r(g − d + r) ≤ d and the “Brill-Noether” locus:

C_{d}^{r} := {D ∈ C^{[d]} | dim(|D|) ≥ r}

is nonempty and has codimension r(d − g + r) in C^{[d]}.

(This is a deep result in Brill-Noether theory...see [ACGH])

The class [C_{d}^{r}] ∈ H^{2r(g−d+r)}(C^{[d]}, Z) has been computed. It is:

[C_{d}^{r}] = (∏_{i=0}^{r} \frac{i!}{(g − d + r + i − 1)!}) ∑_{α=0}^{r}(−1)^{α} \frac{(g − d + r + α − 1)!}{α!(r − α)!}x^{α}θ^{r(g−d+r)−α}

Finally, to compute numbers, one needs to know the intersection numbers of powers of x and of θ. These have also been computed. They are:

∫_{C^{[d]}} x^{d} = 1

∫_{C^{[d]}} θ^{d} = \{ \begin{matrix}0 if d > g \\ \frac{g!}{(g−d)!} if d ≤ g \end{matrix}

From this, for example, one can check that [C_{d}^{r}] is not the zero class(!) by intersecting it with the appropriate power of x.