Abstract
We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). We prove three results about the topology of the twistor discriminant locus of an algebraic surface in \({\mathbb{CP}^3}\). First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.
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References
Altavilla A.: Twistor interpretation of slice regular functions. J. Geom. Phys. 123, 184–208 (2018)
A. Altavilla and E. Ballico, Twistor lines on algebraic surfaces in the complex projective space, Ann Glob Anal Geom (2018), https://doi.org/10.1007/s10455-018-9640-2
A. Altavilla and E. Ballico, Algebraic surfaces with infinitely many twistor lines, e-print arXiv:1902.00010
A. Altavilla and G. Sarfatti, Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in \({\mathbb{CP}^3}\), Math. Z. (2018), https://doi.org/10.1007/s00209-018-2225-8
Armstrong J.: The twistor discriminant locus of the Fermat cubic. New York J. Math. 21, 485–510 (2015)
J. Armstrong, M. Povero, and S. Salamon, Twistor lines on cubic surfaces, Rend. Semin. Mat. Univ. Politec. Torino 71 no. 3–4 (2013), 317–338.
J. Armstrong and S. Salamon, Twistor topology of the Fermat cubic, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), paper 061, 12 pp.
Ballico E.: Conformal automorphisms of algebraic surfaces and algebraic curves in the complex projective space. Journal of Geometry and Physics 134, 153–160 (2018)
M.F. Atiyah, N.J. Hitchin and I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 no. 1711 (1978), 425–461.
J. Bochnak, M. Coste and M.-F. Roy., Real algebraic geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 3, 36, Springer-Verlag, Berlin, 1998. x+430 pp. ISBN: 3-540-64663-9
Chirka E.M.: Orthogonal complex structures in \({\mathbb{R}^4}\). Russ. Math. Surv. 73, 91–159 (2018)
M. Coste, An Introduction to Semialgebraic Geometry, preprint, 2002, available online from http://perso.univ-rennes1.fr/michel.coste/polyens/SAG.pdf.
P. de Bartolomeis and A. Nannicini, Introduction to differential geometry of twistor spaces, in: Geometric theory of singular phenomena in partial differential equations (Cortona, 1995), 91–160, Sympos. Math., XXXVIII, Cambridge Univ. Press, Cambridge, 1998.
I.V. Dolgachev, Classical algebraic geometry. A modern view. Cambridge University Press, Cambridge, 2012. xii+639 pp. ISBN: 978-1-107-01765-8
G. Gentili, S. Salamon, and C. Stoppato, Twistor transforms of quaternionic functions and orthogonal complex structures, J. Eur. Math. Soc. (JEMS) 16 no. 11 (2014), 2323–2353.
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, vol. 14, Springer-Verlag, New York–Heidelberg, 1973. x+209 pp.
P. Griffiths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp. ISBN: 0-471-32792-1
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer- Verlag, New York–Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9
J. Kim, C. LeBrun and M. Pontecorvo, Scalar-flat Kähler surfaces of all genera, J. Reine Angew. Math. 486 (1997), 69–95.
C. LeBrun, Anti-self-dual metrics and Khler geometry, in: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zrich, 1994), 498–507, Birkhäuser, Basel, 1995.
LeBrun C.: Scalar-flat Khler metrics on blown-up ruled surfaces. J. Reine Angew. Math. 420, 161–177 (1991)
C. LeBrun, Explicit self-dual metrics on \({\mathbb{CP}^2{\#}{\ldots}{\#}\mathbb{CP}^2}\), J. Differential Geom. 34 no. 1 (1991), 223–253.
J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, vol. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. iii+122 pp.
Mumford D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9, 5–22 (1961)
M. Pontecorvo, On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 no. 2 (1992), 653–661.
S. Salamon and J. Viaclovsky, Orthogonal complex structures on domains in \({\mathbb{R}^4}\), Math. Ann. 343 no. 4 (2009), 853–899. Also arXiv version arXiv:0704.3422v1.
Shapiro G.: On discrete differential geometry in twistor space. J. Geom. Phys. 68, 81–102 (2013)
G. Xu, Subvarieties of general hypersurfaces in projective space, J. Differential Geom. 39 no. 1 (1994), 139–172.
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GNSAGA of INdAM
MIUR PRIN 2015 “Geometria delle varietà algebriche”
SIR grant “NEWHOLITE – New methods in holomorphic iteration” n. RBSI14CFME, SIR grant AnHyC – Analytic aspects in complex and hypercomplex geometry n. RBSI14DYEB and Fondazione Bruno Kessler–CIRM (Trento) “Research In Pairs” program. Moreover, the present paper was submitted while the first author was a postdoc (assegno di ricerca) at Dipartimento Di Matematica, Università di Roma “Tor Vergata”.
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Altavilla, A., Ballico, E. Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere. Milan J. Math. 87, 57–72 (2019). https://doi.org/10.1007/s00032-019-00292-5
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DOI: https://doi.org/10.1007/s00032-019-00292-5