Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022

Nobuhiro Honda [a] and Ayato Minagawa [a,b]

On the cuspidal locus in the dual varieties of Segre quartic surfaces

Pages: 551-610
Received: 16 December 2021
Accepted: 27 July 2022
Mathematics Subject Classification: 53C26, 14D06.
Keywords: Segre surfaces, cuspidal curves, projectively dual varieties, minitwistor spaces.
Authors address:
[a]: Tokyo Institute of Technology, Department of Mathematics, Meguro, Japan
[b]: Internet Initiative Japan Inc.

The first author was partially supported by JSPS KAKENHI Grant 16H03932

Full Text (PDF)

Abstract: Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.

M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461. MR0506229
W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces, 2nd edition, Springer-Verlag, Berlin, 2004. MR2030225
I. V. Dolgachev, Classical algebraic geometry, A modern view, Cambridge Univ. Press, Cambridge, 2012. MR2964027
N. J. Hitchin, Complex manifolds and Einstein's equations, Lecture Notes in Math., 970, Springer, Berlin-New York, 1982, 73-99. MR0699802 %MR0699802, Zbl 0507.53025.
W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, Vol. II, Cambridge Univ. Press, Cambridge, 1994. MR1288306
N. Honda and F. Nakata, Minitwistor spaces, Severi varieties, and Einstein-Weyl structure, Ann. Global Anal. Geom. 39 (2011), 293-323. MR2769301
N. Honda, Segre quartic surfaces and minitwistor spaces, New York J. Math. 28 (2022), 672-704. MR4407496
R. Penrose, Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31-52. MR0439004
E. Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer-Verlag, Berlin, 2006. MR2247603
A. Tannenbaum, Families of algebraic curves with nodes, Compositio Math. 41 (1980), 107-126. MR0578053
E. A. Tevelev, Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences, 133, Springer-Verlag, Berlin, 2005. MR2113135
C. Voisin, Hodge theory and complex algebraic geometry, II, Cambridge Stud. Adv. Math., 77, Cambridge Univ. Press, Cambridge, 2003. MR1997577
J. M. Wahl, Deformations of plane curves with nodes and cusps, Amer. J. Math. 96 (1974), 529-577. MR0387287

Home Riv.Mat.Univ.Parma