Abstract
In order to obtain existence criteria for orthogonal instanton bundles on \(\mathbb {P}^n\), we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on \(\mathbb {P}^n\) and prove that every orthogonal instanton bundle with no global sections on \(\mathbb {P}^n\) and charge \(c\ge 2\) has rank \(r \le (n-1)c\). We also prove that when the rank r of the bundles reaches the upper bound, \(\mathcal {M}_{\mathbb {P}^n}^{\mathcal {O}}(c,r)\), the coarse moduli space of orthogonal instanton bundles with no global sections on \(\mathbb {P}^n\), with charge \(c\ge 2\) and rank r, is affine, smooth, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in \(\mathcal {M}_{\mathbb {P}^n}^{\mathcal {O}}(c,r)\), whenever is non-empty.
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Acknowledgements
The first author was supported by by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001, Processes Numbers 99999.000282/2016-02 and 88887.308429/2018-00. The second author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Grant 2017/03487-9, by CNPq Grant 303075/2017-1 and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. The third author was partially supported by MTM2016-78623-P.
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Andrade, A.V., Marchesi, S. & Miró-Roig, R.M. Irreducibility of the moduli space of orthogonal instanton bundles on \(\mathbb {P}^n\). Rev Mat Complut 33, 271–294 (2020). https://doi.org/10.1007/s13163-019-00317-y
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DOI: https://doi.org/10.1007/s13163-019-00317-y
Keywords
- Orthogonal instanton bundles
- Symmetric forms
- Moduli spaces
- Geometric invariant theory
- Splitting type
- Kronecker modules