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# Orthogonal Bundles and Skew-Hamiltonian Matrices

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Published:2015-07-21
Printed: Oct 2015
• Roland Abuaf,
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
• Ada Boralevi,
Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 265, 34136 Trieste, Italy
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## Abstract

Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial splitting on the general line, is smooth irreducible of dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and $r=n-1$ and $n\ge 8$. We speculate that the result holds in greater generality.
 Keywords: orthogonal vector bundles, moduli spaces, skew-Hamiltonian matrices
 MSC Classifications: 14J60 - Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 15B99 - None of the above, but in this section

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