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

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

Sukhendu Mehrotra (Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile; and Chennai Mathematical Institute, Kelambakkam, India)

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

Full Text (PDF format)