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Twistor lines on algebraic surfaces

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Abstract

We give quantitative and qualitative results on the family of surfaces in \(\mathbb {CP}^3\) containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterward we prove that twistor lines are Zariski dense in the Grassmannian Gr(2, 4). Then, for any degree \(d\ge 4\), we give lower bounds on the maximum number of twistor lines contained in a degree d surface. The smooth and singular cases are studied as well as the j-invariant one.

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Authors and Affiliations

  1. Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133, Rome, Italy

    A. Altavilla

  2. Dipartimento Di Matematica, Università di Trento, Via Sommarive 14, 38123, Povo, Trento, Italy

    E. Ballico

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  1. A. Altavilla
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  2. E. Ballico
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Correspondence to A. Altavilla.

Additional information

A. Altavilla, E. Ballico: GNSAGA of INdAM. E. Ballico: MIUR PRIN 2015 “Geometria delle varietà algebriche”. A. Altavilla: FIRB 2012 Geometria differenziale e teoria geometrica delle funzioni, SIR Grant “NEWHOLITE - New methods in holomorphic iteration” n. RBSI14CFME and SIR Grant AnHyC - Analytic aspects in complex and hypercomplex geometry n. RBSI14DYEB. The first author wishes to thank also the Clifford Research Group at Ghent University where part of this project was carried out.

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Altavilla, A., Ballico, E. Twistor lines on algebraic surfaces. Ann Glob Anal Geom 55, 555–573 (2019). https://doi.org/10.1007/s10455-018-9640-2

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  • DOI: https://doi.org/10.1007/s10455-018-9640-2

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