Torelli theorems for some Steiner bundles

Appendix A: The theorem of Dolgachev–Kapranov

Let $X \subseteq \textbf{P}(V) = \textbf{P}^r$ be a finite set of $d \ge r+1$ points in linear general position, and denote by $\mathcal {I}= \mathcal {I}_X \subseteq \mathcal {O}_{\textbf{P}(V)}$ the ideal sheaf of X. Green shows in [14, Theorem (3.c.6)] that $K_{r-2, 2}(\textbf{P}^r, \mathcal {I}; V) \ne 0$ if and only if X lies on a rational normal curve.¹ Given the arguments from the previous section, it is natural to expect that one can use this to get a new proof of the Torelli-type theorem of Dolgachev and Kapranov from [8] (along with the numerical improvements by Vallès [18]). Inspired by some of the techniques in [14], we indicate here how this goes. For simplicity we assume that $r \ge 3$.

Note to begin with that each $H_{*}^{{i}} \big ( {\textbf{P}(V)} , {\mathcal {I}}\big ) =_{\text {def}} \oplus _k\, H^{{i}} \big ( {\textbf{P}(V)} , {\mathcal {I}(k)}\big )$ is a graded module over the symmetric algebra $\text {Sym}(V)$. In particular, there is a natural map

$$\begin{aligned} H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}}\big ) \otimes V \longrightarrow H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}(1)}\big ) \end{aligned}$$

(A.1)

On the other hand, every point $x \in X\subseteq \textbf{P}(V)$ determines a dual hyperplane , and so X itself gives rise to a normal crossing hyperplane arrangement $\Sigma H_i$ on . One checks that the Dolgachev–Kapranov bundle is the Steiner bundle on determined by the multiplication map

$$\begin{aligned} H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}(1)}\big ) ^* \otimes V \longrightarrow H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}}\big ) ^*, \end{aligned}$$

deduced from (A.1). It is elementary that $X \subseteq \text {Vall}\grave{\text {e}}\text {s}(E)$, and we want to verify

Proposition A.1

If $X \subsetneqq \text {Vall}\grave{\text {e}}\text {s}(E)$, then X lies on a rational normal curve in $\textbf{P}({V})$.

Equivalently, fix a subspace $W \subseteq V$ of codimension one that generates $\mathcal {O}_X$. In view of Lemma 1.2, we need to show that if the mapping

$$\begin{aligned} H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}_X(1)}\big ) ^* \otimes W \longrightarrow H^{{1}} \big ( {\textbf{P}(V)} , {\mathcal {I}_X}\big ) ^* \end{aligned}$$

(A.2)

fails to be surjective, then X lies on a rational normal curve.

Since $W \subseteq V$, each $H_{*}^{{i}} \big ( {\textbf{P}(V)} , {\mathcal {I}}\big )$ has the structure of a $\text {Sym}(W)$-module. The first point is that in bounded degrees, one can realize these as the cohomology modules of a sheaf $\mathcal {J}$ on $\textbf{P}(W)$. Specifically:

Lemma A.2

For any suitably large integer $k_0 \gg 0$ one can construct a coherent sheaf $\mathcal {J}$ on $\textbf{P}(W)$ with the property that for $i < r-1 = \dim \textbf{P}(W)$ there are isomorphisms

$$\begin{aligned} H_{*}^{{i}} \big ( {\textbf{P}(V)} , {\mathcal {I}}\big ) _{\le k_0} \ \cong \ H_{*}^{{i}} \big ( {\textbf{P}(W)} , {\mathcal {J}}\big ) _{\le k_0} \end{aligned}$$

in degrees $\le k_0$, and these isomorphisms are compatible with the $\text {Sym}(W)$-module structures on both sides.

Granting the Lemma for the time being, we give the

Proof of Proposition A.1

Tensoring the universal Koszul complex on $\textbf{P}(W)$ by $\mathcal {J}$, one arrives at a long exact sequence

$$\begin{aligned} 0 \longrightarrow \Lambda ^r W \otimes \mathcal {J}\longrightarrow \Lambda ^{r-1}W \otimes \mathcal {J}(1) \longrightarrow \Lambda ^{r-2}W \otimes \mathcal {J}(2) \longrightarrow \ldots \longrightarrow \mathcal {J}(r)\longrightarrow 0 \end{aligned}$$

of sheaves on $\textbf{P}(W)$. This in turn gives rise to a hypercohomology spectral sequence abutting to zero. The bottom two rows of its $E_1$ page have the form

where the cohomology groups are taken on $\textbf{P}(W)$. The assumption (A.2) means (thanks to the Lemma) that the map

$$\begin{aligned} H^1(\mathcal {I}) \, = \, \Lambda ^r W \otimes H^1(\mathcal {J}) \longrightarrow \Lambda ^{r-1} W \otimes H^1(\mathcal {J}(1)) \, = \, W^* \otimes H^1(\mathcal {I}(1)) \end{aligned}$$

has a non-trivial kernel. This must cancel against the $E_2$ term coming from the bottom row of the spectral sequence. In other words,

$$\begin{aligned} K_{r-2,2}(\textbf{P}(W), \mathcal {J}) \ \ne \ 0. \end{aligned}$$

But $K_{r-2,1}(\textbf{P}(W), \mathcal {J}) = 0$ since X does not lie on a hyperplane, so by (the analogue of) Lemma 2.1 (ii), we conclude that $K_{r-2,2}(\textbf{P}(V), \mathcal {I}) \ne 0$. Then Green’s theorem applies to put X on a rational normal curve. $\square$

Proof of Lemma A.2

Let $w \in \textbf{P}(V)$ be the point corresponding to $W \subseteq V$, so that in particular $w \not \in X$. Projection $\pi : \big ( \textbf{P}(V) - \{ w \}\big ) \longrightarrow \textbf{P}(W)$ from w gives an identification

$$\begin{aligned} \textbf{P}(V) - \{ w \} \ \cong \ \text {Spec}_{\textbf{P}(W)}\Big (\text {Sym}\big ( \mathcal {O}_{\textbf{P}(W)} \oplus \mathcal {O}_{\textbf{P}(W)}(-1) \big ) \Big ). \end{aligned}$$

Then for $k_0 \gg 0$, one arrives at a surjective mapping

$$\begin{aligned} \varepsilon : \text {Sym}^{k_0} \big ( \mathcal {O}_{\textbf{P}(W)} \oplus \mathcal {O}_{\textbf{P}(W)}(-1) \big ) \longrightarrow \pi _* \mathcal {O}_X \end{aligned}$$

of sheaves on $\textbf{P}(W)$. It suffices to take $\mathcal {J}= \ker (\varepsilon )$. $\square$