tags: moduli, Abelian varieties

Functors \(Pic\) and \(Div\)

We will denote base change with a subscript: \(X_T = X \times T\).

If \(X\) is a scheme, the Picard group of \(X\) is defined to be the group of isomorphism classes of invertible sheaves on \(X\). The relative Picard functor of an \(S\)-scheme \(X\) is defined as \[ \operatorname{Pic}_{X/S}(T) := \operatorname{Pic}(X_T) / \operatorname{Pic}(T) \] where the embedding \(\operatorname{Pic}(T) \hookrightarrow \operatorname{Pic}(X_T)\) is given by the pullback along the structure maps \(X_T \to T\).

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless \(X \to S\) is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If \(f: X \to S\) is a morphism of schemes then a relative effective divisor on \(X\) is an effective divisor \(D\) such that \(D\) is flat over \(S\). For a morphism \(X \to S\) define the functor of relative divisors \[ \operatorname{Div}_{X/S}(T) := \{\ \textrm{ relative effective divisors on } X_T/T \ \} \]

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let \(X \to S\) be a flat morphism. Let \(D\) be a closed subscheme of \(X\) flat over \(S\). Then \(D\) is relative effective divisor in a neighbourhood of \(x \in X\) if and only if \(D_s\) is cut out in a neighbourhood of \(x\) in \(X_s\), where \(s\) is the image of \(x\), by a single non-zero element of \(\operatorname{\mathcal{O}}_{X_s,x}\) (which amounts to being an effective divisor in \(X_s\) but we won’t prove it).

Proof. Left to right. Multiplication by the element \(f\) that cuts out \(D\) induces the short exact sequence of \(\operatorname{\mathcal{O}}_{S,s}\)-modules \[ 0 \to \operatorname{\mathcal{O}}_{X,x} \to \operatorname{\mathcal{O}}_{X,x} \to \operatorname{\mathcal{O}}_{D,x} \to 0 \] Tensoring it with \(k(s)\) we get \[ \begin{array}{c} \ldots \to \operatorname{Tor}_1(\operatorname{\mathcal{O}}_{X,x}, k(s)) \to \operatorname{Tor}_1(\operatorname{\mathcal{O}}_{X,x}, k(s)) \to \operatorname{Tor}_1(\operatorname{\mathcal{O}}_{D,x}, k(s)) \to \\ \to \operatorname{\mathcal{O}}_{X,x} \otimes k(s) \to \operatorname{\mathcal{O}}_{X,x} \otimes k(s) \to \operatorname{\mathcal{O}}_{D,x} \otimes k(s) \to 0\\ \end{array} \] Since \(D\) is flat in a neighbourhood of \(x\), the third term \(\operatorname{Tor}_1(\operatorname{\mathcal{O}}_{D,x}, k(s))\) vanishes. Therefore, the element \(f \otimes k(s)\) cuts out \(D_s\) in \(X_s\), and \(D_s\) is an effective divisor in \(X_s\).

Right to left: suppose \(D_s\) is locally cut out by non-zero divisor of \(\operatorname{\mathcal{O}}_{X_s,x}\) then we have to show \(D\) is locally cut out by a single element of \(\operatorname{\mathcal{O}}_{X,x}\).

Consider the exact secquence \[ 0 \to I_{D,x} \to \operatorname{\mathcal{O}}_{X,x} \to \operatorname{\mathcal{O}}_{D,x} \to 0 \] then the long exact sequence associated to tensoring with \(k(s)\) is \[ \operatorname{Tor}_1(\operatorname{\mathcal{O}}_{X,x}, k(s)) \to \operatorname{Tor}_1(\operatorname{\mathcal{O}}_{D,x}, k(s)) \to I_{D,x} \otimes k(s) \to \operatorname{\mathcal{O}}_{X,x} \otimes k(s) \to \operatorname{\mathcal{O}}_{D,x} \otimes k(s) \to 0 \] as \(I_{D,x} = f\operatorname{\mathcal{O}}_{X,x}\) and \(f\) is regular in the fibre \(X_s\), \(I_{D,x} \otimes k(s) \to \operatorname{\mathcal{O}}_{X,x} \otimes k(s)\) is multiplication by \(f \otimes k(s)\), so in fact is an inclusion. \(\operatorname{Tor}_1(\operatorname{\mathcal{O}}_{X,x}, k(s))\) vanishies sinche \(X\) is flat over \(S\), and therefore, finally, \(\operatorname{Tor}_1(\operatorname{\mathcal{O}}_{D,x}, k(s))=0\), and \(D\) is flat over \(S\).

Since, as it follows from the premise, the multplication by \(f \otimes k(s)\) induces an isomorphism \(\operatorname{\mathcal{O}}_{X,x} \otimes k(s) \to I_{D,x} \otimes k(s)\), it follows by Nakayama’s lemma that the kernel of the multiplication by \(f\) is trivial, and so \(f\) is not a zero divisor.

Corollary. \(\operatorname{Div}\) is representable by an open subscheme of \(\operatorname{Hilb}\).

Proof. Let \(W \subset X \times \operatorname{Hilb}_{X/S}\) be the universal scheme. Then the property of being an effective divisor on \(X \times \operatorname{Hilb}\) in a neighourhood of a given point is an open property because of the previous lemma, let \(U \subset W\) be the set of such points. Then since projection to \(\operatorname{Hilb}_{X/S}\) is proper, the image of \(U\) in \(\operatorname{Hilb}_{X/S}\) is open. Let us show that it represents \(\operatorname{Div}_{X/S}\). Note that for any \(y \in O\), \(W_y\) is an effective divisor in \(X_y\) by the Lemma above.

Given relative effective divisor \(D \subset X_T\) there exists a map \(\iota_D: T \to \operatorname{Hilb}_{X/S}\) such that \(D = W \times_{\operatorname{Hilb},\iota_D} T\). At every point \(t \in \operatorname{Im}(\iota_D: T \to \operatorname{Hilb})\), the closed subscheme \(W_t\) is a divisor in \(X_t\), and by the Lemma above \(W_T\) is a relative effective divisor.

representability of \(\operatorname{Pic}(X/S)\), Mumford’s method

Suffices to represent \(\operatorname{Pic}^\tau\), the component of \(\operatorname{Pic}\) that contains numerically trivial divisors, because components corresponding to different numerical classes are isomorphic. In fact, it will be more convenient to fix a very ample divisor \(\xi\) and represent the compononet \(\operatorname{Pic}^\xi\) of \(\operatorname{Pic}\) consisting of divisors numerically equivalent to \(\xi\). We pick \(\xi\) to be 0-regular (regularity was described in the previous post).

There is a little semi-tautological argument that it is enough to show representability of \(\operatorname{Pic}^\xi\) (given that \(\operatorname{Div}^\xi\) is representable).

We consider the morphism of fuctors \(\Phi: \operatorname{Div}\to \operatorname{Pic}\) and then restrict it to \(\operatorname{Pic}^\xi\). The goal is to construct a section \(s: \operatorname{Pic}^\xi \to \operatorname{Div}^\xi\).

Lemma. Assume \(s\) exists, then \(\operatorname{Pic}\) is representable.

Proof. The morphism \(s \circ \Phi\) is an endomorphism of \(\operatorname{Div}^\xi\), which is representable, say by an \(S\)-scheme \(D\). Then \(s \circ \Phi(\operatorname{id}_D)\) is an endomorphism \(f\) of the scheme \(D\). Consider the fibre product \(D \times_{\Delta, D \times D, \operatorname{id}\times f} D\), where \(\Delta: D \to D \times_S D\) is the diagonal map. Call this fibre product \(P\).

We claim that \(P\) represents \(\operatorname{Pic}\).

Indeed, \(\operatorname{Hom}(T, P)\) is isomorphic, by construction, to the set of pairs of morphsims \(\alpha, \beta: T \to D\) such that \(\Delta(\alpha) = \operatorname{id}\times f (\beta)\), i.e. \(\operatorname{Hom}(T, P)\) is the image of \(\operatorname{Hom}(T, D)\) under \(s \circ \Phi\). This means that \(P\) represents \(\operatorname{Pic}^\xi\). \(\square\)

The section is constructed as follows (Mumford assumes \(S=k\), a field, not sure how this is restrictive; on the other hand he also assumes \(X\) is a surface, this seems to be crucial later).

Suppose we are given an invertible sheaf \(\operatorname{\mathcal{L}}\) no \(X \times T\). Denote \(\operatorname{\mathcal{M}}_x\) its restriction to \({x} \times T\), which can be considered as a sheaf on \(T\), and let \(\operatorname{\mathcal{E}}\) be the sheaf of global sections \((p_T)_* \operatorname{\mathcal{L}}\).

We pick a finite number of points \(x_1, \ldots, x_N \in X\), and consider the natural morphism \[ h: \operatorname{\mathcal{E}}\to \bigoplus_{i=1}^N \operatorname{\mathcal{M}}_{x_i}, s \mapsto (s(x_1), \ldots, s(x_N)) \] Assuming the rank of \(\operatorname{\mathcal{E}}\) is \(r\) (and this is the same for invertible sheaves of the same numerical class, if this class is sufficiently ample), we can wedge this \(r-1\) times to get a homomorphism of invertible sheaves \[ (\wedge h)^*: \otimes \operatorname{\mathcal{M}}_{x_i} \to \operatorname{Hom}(\wedge \operatorname{\mathcal{E}}, \operatorname{\mathcal{O}}_T) \] This gives a canonical morphism of sheaves \[ \operatorname{\mathcal{O}}_T \to \operatorname{\mathcal{E}}\otimes ((\wedge \operatorname{\mathcal{E}})^{-1} \otimes (\otimes M_{x_i})) \] and hence a canonical section \[ \sigma \in H^0(X \times T, \operatorname{\mathcal{L}}\otimes (p_T)^* ((\wedge \operatorname{\mathcal{E}})^{-1} \otimes (\otimes M_{x_i})) \] Using some magick related to regularity (takes part in the choice of \(\xi\)), one can show that the zero locus of this section does not vanish on any fibre \(X_t\).

(All this is Lectures 19-20 in ``Lectures on curves on algebraic surface’’)

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