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Automorphisms of relative Quot schemes. (English) Zbl 1423.14027

Summary: Let \(X\rightarrow S\) be a smooth family of projective curves over an algebraically closed field \(k\) of characteristic zero. Assume that both \(X\) and \(S\) are smooth projective varieties and let \(E\) be a vector bundle of rank \(r\) over \(X\) and \(\mathbb{P}(E)\) be its projectivization. Fix \(d\ge 1\). Let \(\mathcal{Q}(E,d)\) be the relative Quot scheme of torsion quotients of \(E\) of degree \(d\). Then we show that if \(r\ge 3\), then the identity component of the group of automorphisms of \(\mathcal{Q}(E,d)\) over \(S\) is isomorphic to the identity component of the group of automorphisms of \(\mathbb{P}(E)\) over \(S\). We also show that under additional hypotheses, the same statement is true in the case when \(r=2\). As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of \(\mathcal{O}^r_C,\) where \(r\ge 3\) and \(C\) a smooth projective curve of genus \(\ge 2\) is computed.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14J50 Automorphisms of surfaces and higher-dimensional varieties
14M17 Homogeneous spaces and generalizations
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References:

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