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On the cotangent sheaf of quot-schemes. (English) Zbl 0934.14008

Let \(f: X \rightarrow S\) be a smooth projective morphism of Noetherian schemes of relative dimension \(d\). Let \({\mathcal O}_X(1)\) be a relatively very ample line bundle and \(\omega_{X/S}\) be the relative dualising sheaf on \(X\). Let \(H\) be a coherent sheaf on \(X\). Let \(Q\) denote the relative Grothendieck Quot-scheme of flat quotients of \(H\) with a fixed Hilbert polynomial \(P\) with respect to \({\mathcal O}_X(1)\). Let \(p\) and \(q\) denote the projections from \(Q \times_S X\) to \(Q\) and \(X\), respectively. Let \(K\) and \(F\) be respectively the universal subsheaf and quotient sheaf on \(Q\times_S X\). The author has the following global description of the relative dualising sheaf \(\Omega _{Q/S}\).
Theorem 1. There is a natural isomorphism of \({\mathcal O}_Q\)-sheaves \[ h : {\mathcal E}xt^d_p(F, K\otimes q^*\omega_{X/S}) \rightarrow \Omega_{Q/S}. \] A useful application of this result is a similar description of the cotangent sheaf of the moduli space \(M_{X/S}\) of stable sheaves on the fibres of \(f\). Assume further that \(f\) has geometrically integral fibres and \(S\) is of finite type over a field of characteristic 0. The moduli space is the quotient of an open subscheme \(R\) of a suitable Quot-scheme by \(PGL(N)\). Let \(F_R\) denote the universal quotient bundle on \(R\) and \(\pi: R \rightarrow M\) the canonical morphism.
Theorem 2. There is a natural \(GL(N)\)-equivariant isomorphism \[ h': {\mathcal E}xt^{d-1}_p(F_R, F_R \otimes \omega) \rightarrow\pi^*\Omega_{M/S}. \] Since the centre acts trivially, the sheaves \({\mathcal E}xt^i_p(F_R, F_R\otimes \omega)\) descend to coherent sheaves \(E^i\) on \(M\). Theorem 2 implies that \(E^{d-1} \approx \Omega_{M/S}\). In particular, if there is a universal family \(F\) on \(M \times_S X\) then \({\mathcal E}xt^{d-1}_p (F, F\otimes \omega) \approx \Omega_{M/S}\).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C05 Parametrization (Chow and Hilbert schemes)
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References:

[1] Maruyama M., J. Math. Kyoto Univ. 18 pp 557– (1978)
[2] DOI: 10.1007/BF01389137 · Zbl 0565.14002 · doi:10.1007/BF01389137
[3] DOI: 10.1007/BF02698887 · Zbl 0891.14005 · doi:10.1007/BF02698887
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