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# Advances in Theoretical and Mathematical Physics

## Volume 20 (2016)

### Number 6

### Automorphisms of the generalized quot schemes

Pages: 1473 – 1484

DOI: http://dx.doi.org/10.4310/ATMP.2016.v20.n6.a6

#### Authors

#### Abstract

Given a compact connected Riemann surface $X$ of genus $g \geq 2$, and integers $r \geq 2$, $d_p \gt 0$ and $d_z \gt 0$, in [BDHW], a generalized quot scheme $\mathcal{Q}_X (r, d_p, d_z)$ was introduced. Our aim here is to compute the holomorphic automorphism group of $\mathcal{Q}_X (r, d_p, d_z)$. It is shown that the connected component of $\mathrm{Aut} \; \mathcal{Q}_X (r, d_p, d_z)$ containing the identity automorphism is $\mathrm{PGL}(r, \mathbb{C})$. As an application of it, we prove that if the generalized quot schemes of two Riemann surfaces are holomorphically isomorphic, then the two Riemann surfaces themselves are isomorphic.

#### Keywords

generalized quot scheme, vector fields, automorphism group, symmetric product

#### 2010 Mathematics Subject Classification

14D21, 14D23, 14H60