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Stability of tautological vector bundles on Hilbert squares of surfaces. (English) Zbl 1208.14036

Let \(Z\subset S \times \mathrm{Hilb}^2(S)\) be the universal subscheme for the Hilbert square of a smooth projective surface \(S\), parameterizing the pairs \((x,\xi) \in S \times \mathrm{Hilb}^2(S)\) with \(x \subset \xi\), and let \(p: Z \rightarrow S\) and \(q: Z \rightarrow \mathrm{Hilb}^2(S)\) be the projections. Any line bundle \(L \rightarrow S\) defines a rank two vector bundle \(L^{[2]} := q_*p^*L \rightarrow \mathrm{Hilb}^2(S)\), called the tautological bundle for \(L\). The author proves the following
Theorem. If \(h^0(S,L) \geq 2\) then for \(N >>0\) the tautological bundle \(L^{[2]}\) is \(\mu\)-stable for the polarizations \(H_N\) given by \(Sym^2(NH) - E\), where \(H\) is an ample divisor on \(S\) and \(E \subset \mathrm{Hilb}^2(S)\) is the exceptional divisor of the Hilbert-Chow morphism \(\mathrm{Hilb}^2(S) \rightarrow \mathrm{Sym}^2(S)\).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14C05 Parametrization (Chow and Hilbert schemes)
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References:

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