Deformation of rank 2 quasi-bundles and some strange dualities for rational surfaces
Let \({\mathcal M}_n\) (resp., \({\mathcal N}_d\)) be the coarse moduli space of rank 2 torsion free (resp., pure dimension 1) Gieseker-Maruyama semistable sheaves on the projective plane \({\mathbb P}^2\) with trivial determinant and \(c_2 = n \geq 0\) (resp., with determinant \({\mathcal O}_{{\mathbb P}^2}(d)\), \(d \geq 1\), and \(\chi = 0\)). Let \(\rho : {\mathcal N}_d \rightarrow | \, {\mathcal O}_{{\mathbb P}^2}(d)\, |\) be the morphism associating to \([{\mathcal G}] \in {\mathcal N}_d\) its scheme theoretic support (defined by the determinant of a locally free presentation of \(\mathcal G\)). Using the simple fact asserting that if \(f : X \rightarrow T\) is a projective morphism and \(\mathcal F\) is a coherent sheaf on \(X\), flat over \(T\), with \(\text{Supp}\, {\mathcal F}\) of relative dimension 1 over \(T\), and with \(\chi ({\mathcal F}_t) = 0\), \(\forall t \in T\), then \([\text{det}(\text{R}f_{\ast}{\mathcal F})]^{-1}\) has a global section vanishing exactly on \(\{ t\in T \, | \, \text{H}^0({\mathcal F}_t) \neq 0 \}\), one can show that there exists a line bundle \({\mathcal D}_d\) (resp., \({\mathcal R}_n\)) on \({\mathcal M}_n\) (resp., \({\mathcal N}_d\)) and a global section of the line bundle \({\mathcal D}_d \boxtimes {\mathcal R}_n\) on \({\mathcal M}_n \times {\mathcal N}_d\) whose zero locus is \(\{ ([{\mathcal F}],[{\mathcal G}])\, | \, \text{H}^0({\mathbb P}^2,{\mathcal F} \otimes {\mathcal G}) \neq 0 \}\). This section gives rise to the \textit{strange-duality} map: \[ (\text{SD})_{n,d} : \text{H}^0({\mathcal N}_d,{\mathcal R}_n)^{\vee} \longrightarrow \text{H}^0({\mathcal M}_n,{\mathcal D}_d)\, . \] J. Le Potier conjectured that \((\text{SD})_{n,d}\) is an isomorphism. Now, for \(d = 1\) or 2, \(\rho\) is an isomorphism and \({\mathcal R}_n \simeq {\rho}^{\ast}{\mathcal O}(n)\) (for example, the points of \({\mathcal N}_1\) are of the form \([{\mathcal O}_L(-1)]\) with \(L\) line in \({\mathbb P}^2\)). \textit{G. Danila} [Ann. Inst. Fourier (Grenoble) 50, No. 5, 1323--1374 (2000; Zbl 0952.14010); Bull. Soc. Math. Fr. 130, No. 1, 1--33 (2002; Zbl 1038.14004)] computed the dimension of \(\text{H}^0({\mathcal N}_3, {\mathcal R}_n)\), showed that \((\text{SD})_{n,d}\) is injective for \(d \leq 3\) and proved that \((\text{SD})_{n,d}\) is an isomorphism for \(d = 1\) and \(n \leq 19\), and for \(d = 2\) or 3 and \(n \leq 5\). In the paper under review, the author proves that \((\text{SD})_{n,d}\) is an isomorphism for \(d = 1\) or 2 and \(n\) arbitrary using different, more natural arguments. For technical reasons, he works with the moduli stack \(M_n\) of \(\mu\)-\textit{semistable} rank 2 \textit{quasi-bundles} on \({\mathbb P}^2\) with trivial determinant and \(c_2 = n\). A quasi-bundle on \({\mathbb P}^2\) is a torsion free sheaf \(\mathcal F\) such that \({\mathcal F}^{\vee \vee}/{\mathcal F} \simeq {\mathcal O}_Z\), where \(Z\) is a reduced 0-dimensional subscheme of \({\mathbb P}^2\). \(M_n\) is an open substack of the stack \(\mathfrak{M}^{\mu}_n\) of \(\mu\)-semistable rank 2 sheaves on \({\mathbb P}^2\) with the above invariants. \(\mathfrak{M}^{\text{GM}}_n\) (selfexplanatory notation) is a ``large'' open substack of \(\mathfrak{M}^{\mu}_n\), there exists a line bundle \({\mathcal D}^{\mu}_d\) on \(\mathfrak{M}^{\mu}_n\) whose restriction to \(\mathfrak{M}^{\text{GM}}_n\) is the pull-back of \({\mathcal D}_d\) via the morphism \(\mathfrak{M}^{\text{GM}}_n \rightarrow {\mathcal M}_n\), and there exists a strange-duality map \[ (\text{SD})^{\mu}_{n,d} : \text{H}^0({\mathcal N}_d,{\mathcal R}_n)^{\vee} \longrightarrow \text{H}^0(\mathfrak{M}^{\mu}_n,{\mathcal D}^{\mu}_d)\, . \] If \({\mathcal F} \in M_n(\text{Spec}\, k)\), the author defines an \(m\)-point \textit{pinch structure} on \(\mathcal F\) to be an ordered collection \(w_1,\dots, w_m\) of distinct points of \({\mathbb P}^2 \setminus \text{Sing}\, {\mathcal F}\) and of epimorphisms \({\mathcal F}(w_i) \rightarrow k\), \(i = 1, \dots ,m\). The kernel \({\mathcal F}^{\prime}\) of the composed morphism \({\mathcal F} \rightarrow {\mathcal F}(w_m) \rightarrow k\) belongs to \(M_{n+1}(\text{Spec}\, k)\) and is endowed with an \((m-1)\)-point pinch structure. Denoting by \(P_mM_n\) the moduli stack of \(\mu\)-semistable rank 2 quasi-bundles on \({\mathbb P}^2\) with trivial determinant and \(c_2 = n\), endowed with an \(m\)-point pinch structure, one gets a morphism of stacks \(P_mM_n \rightarrow P_{m-1}M_{n+1}\). By studying the \textit{infinitesimal deformations of quasi-bundles}, the author shows that \(P_mM_n\) is the normalization of the divisor \(P_{m-1}S^{\ast}_{n+1}\) on \(P_{m-1}M_{n+1}\) corresponding to singular quasi-bundles. There are some naturally defined line bundles on \(P_mM_n\) and it is easy to determine their pull-backs via \(P_mM_n \rightarrow P_{m-1}M_{n+1}\). Moreover, the author expresses \({\mathcal O}_{P_mM_n}(P_mS^{\ast}_n)\) in terms of these line bundles. Consider, now, the sequence of morphisms: \[ P_nM_0 \rightarrow P_{n-1}M_1 \rightarrow \cdots \rightarrow P_iM_{n-i} \rightarrow \cdots \rightarrow M_n \] and let \({\mathcal L}_i\) be the pull-back of \({\mathcal D}^{\mu}_d\, | \, M_n\) to \(P_iM_{n-i}\). Since \(M_0(\text{Spec}\, k)\) consists of sheaves isomorphic to \({\mathcal O}_{{\mathbb P}^2}^{\oplus 2}\), one sees easily that \(P_nM_0\) is isomorphic to the quotient stack \(\text{GL}(2)\setminus (({\mathbb P}^2)^{<n>} \times ({\mathbb P}^1)^n)\), where \(({\mathbb P}^2)^{<n>} = ({\mathbb P}^2)^n \setminus\) the diagonals and where \(\text{GL}(2)\) acts diagonally on \(({\mathbb P}^1)^n\). Using the essential fact that the moduli stack \(M_p\) is \textit{irreducible}, \(\forall p \geq 1\), the author shows that, for \(d = 1\) or 2, the maps \(\text{H}^0(P_iM_{n-i},{\mathcal L}_i) \rightarrow \text{H}^0(P_{i+1}M_{n-i-1}, {\mathcal L}_{i+1})\) are injective (one needs the technical condition \(\text{H}^0({\mathcal O}_{{\mathbb P}^2}(d) \otimes {\omega}_{{\mathbb P}^2}^{\otimes l}) = 0\), \(\forall l \geq 1\)). Moreover, in these two cases, the image of the map \[ \text{H}^0(M_n,{\mathcal D}^{\mu}_d\, | \, M_n) \rightarrow \text{H}^0(P_nM_0,{\mathcal L}_n) \simeq \text{H}^0({\mathcal O}_{{\mathbb P}^2}(d))^{\otimes n} \] is contained in \(\text{Sym}^n\text{H}^0({\mathcal O}_{{\mathbb P}^2}(d))\). Finally, the author checks that the image of the composed morphism: \[ \text{H}^0({\mathcal N}_d,{\rho}^{\ast}{\mathcal O}_{| {\mathcal O}_{\mathbb P}(d) |} (n))^{\vee} \rightarrow \text{H}^0(M_n,{\mathcal D}^{\mu}_d\, | \, M_n) \rightarrow \text{H}^0(P_nM_0,{\mathcal L}_n) \] is exactly \(\text{Sym}^n\text{H}^0({\mathcal O}_{{\mathbb P}^2}(d))\). The method used by the author works not only for \({\mathbb P}^2\) but also for some other rational surfaces.
scientific article; zbMATH DE number 5834476
scientific article