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On the quadratic invariant of binary sextics

Published online by Cambridge University Press:  28 July 2016

MACIEJ DUNAJSKI
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA. e-mail: M.Dunajski@damtp.cam.ac.uk
ROGER PENROSE
Affiliation:
The Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG. e-mail: penroad@wadh.ox.ac.uk

Abstract

We provide a geometric characterisation of binary sextics with vanishing quadratic invariant.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Bryant, R. L. Metrics with exceptional holonomy. Ann. of Math. 126 (1987), 525576.Google Scholar
[2] Dolgachev, I. Lectures on Invariant Theory (Cambridge University Press, 2003).Google Scholar
[3] Doubrov, B. and Dunajski, M. Co–calibrated G 2 structure from cuspidal cubics. Ann. Global Anal. Geom. 42 (2012), 247265.Google Scholar
[4] Dunajski, M. and Godliński, M. GL(2, ℝ) structures, G 2 geometry and twistor theory. Quart. J. Math 63 (2012), 101132.Google Scholar
[5] Dunajski, M. and Sokolov, V. V. On 7th order ODE with submaximal symmetry. J. Geom. Phys. 61 (2011), 12581262.Google Scholar
[6] Eastwood, M. G. and Isaev, A. V. Extracting invariants of isolated hypersurface singularities from their moduli algebras. Math. Ann. 356 (2013), 7398.Google Scholar
[7] Elliott, E. B. An Introduction to the Algebra of Quantics (Oxford University Press, Clarendon Press, Oxford, 1895).Google Scholar
[8] Grace, J. H. and Young, A. The Algebra of Invariants (Cambridge University Press, Cambridge, 1903).Google Scholar
[9] Hitchin, N. Vector Bundles and the Icosahedron. Contemp. Math. 522 Amer. Math. Soc. (Providence, RI 2010), 7187.CrossRefGoogle Scholar
[10] Igusa, J. I. Arithmetic variety of moduli of genus two. Ann. of Math. 72 (1960), 612649.Google Scholar
[11] Kodaira, K. On stability of compact submanifolds of complex manifolds. Amer. J. Math. 85 (1963), 7994.Google Scholar
[12] Kung, J. P. S. and Rota, G. The invariant theory of binary forms. Bull. AMS 10 (1984), 2785.Google Scholar
[13] Mumford, D., Fogarty, J. and Kirwan, F. Geometric Invariant Theory (Springer-Verlag, 1994).CrossRefGoogle Scholar
[14] Olver, P. Classical Invariant Theory (Cambridge University Press, Cambridge, 1999).Google Scholar
[15] Penrose, R. Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 3152.Google Scholar
[16] Penrose, R. Orthogonality of general spin states. Twistor Newsletter 36 (1993).Google Scholar
[17] Penrose, R. and Rindler, W. Spinors and space-time. Two-spinor calculus and relativistic fields. Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1987, 1988).Google Scholar
[18] Sylvester, J. J. On the calculus of forms, otherwise the theory of invariants. Cambridge and Dublin Mathematical Journal IX (1854), 85103.Google Scholar
[19] Sylvester, J. J. On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, with three appendices. Amer. Journ. Math. I. (1878), 64125.Google Scholar
[20] Zimba, J. and Penrose, R. On Bell nonlocality without probabilities: more curious geometry. Stud. Hist. Philos. Sci. 24 (1993), 697720.Google Scholar