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Generic strange duality for \(K3\) surfaces. (English) Zbl 1275.14037

Let \((X,H)\) be a smooth complex polarized projective surface, \(v\) a class in \(K_0(X)\) and \(v^{\perp} \subset K_0(X)\) its orthogonal complement. Consider \(M_v\) the moduli space of (Gieseker) \(H\)-semistable sheaves on \(X\) with class \(v\). Suppose that \(v\) and \(H\) are such that a semistable sheaf is stable, and consider the map \(\Theta: v^{\perp} \to {\mathrm{Pic}}(M_v)\) described via the Fourier–Mukai transform whose kernel is the universal object of the moduli problem. Let \(\Theta_w\) be the image of a vector \(w\) via \(\Theta\).
Choose \(w\) in \(v^{\perp}\). For \(E\) (resp. \(F\)) stable sheaf with class \(v\) (resp \(w\)), suppose that \(H^0(E \boxtimes F)\) and \({\mathrm{Tor}}^i(E,F)\) are trivial for \(i=1,2\) away from codimension \(2\) in \(M_v \times M_w\). Under these conditions, one has a well defined map \({\mathbf{D}}: H^0(M_v, \Theta_w)^* \to H^0(M_w,\Theta_v)\). Strange duality holds if \(\mathbf{D}\) is an isomorphism.
In this paper, the authors consider a primitively polarized \(K3\) surface \((X,H)\), and orthogonal vectors \(v\) and \(w\) with rank \(r\geq 2\) and \(s\geq 3\) respectively. They show that strange duality holds under the further assumptions that \(c_1(v)=c_1(w)=H\), that \(v\) and \(w\) have nonpositive Euler characteristic, and under some dimensional bound on \(M_v\) and \(M_w\) depending on the respective ranks \(r\) and \(s\). If \(r=s=2\), strange duality is shown to hold without any dimensional assumption.
The main result is obtained restricting first to the case where \(X\) has a structure of an elliptic fibration with a section. Strange duality is shown to hold here for \(v\) and \(w\) of rank at least 2, fiber-degree 1, under some bound on the sum of the dimensions of \(M_v\) and \(M_w\) depending on the ranks. Under these assumptions it is also shown that a generic stable sheaf \(E\) in \(M_v\) defines a Theta divisor \(\Theta_E\) on \(M_w\). These two results use the existence of a birational map between \(M_v\) (resp. \(M_w\)) and the Hilbert scheme (of the same dimension of \(M_v\) (resp. \(M_w\))) of points on \(X\). The main statement is obtained from the elliptic case by deformation techniques.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

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