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15-Nodal Quartic Surfaces. Part I: Quintic del Pezzo Surfaces and Congruences of Lines in P3

Published online by Cambridge University Press:  25 October 2022

By
Igor V. Dolgachev
Edited by
Hamid Abban ,
Gavin Brown ,
Alexander Kasprzyk  and
Shigefumi Mori
Hamid Abban
Affiliation:
Loughborough University
Gavin Brown
Affiliation:
University of Warwick
Alexander Kasprzyk
Affiliation:
University of Nottingham
Shigefumi Mori
Affiliation:
Kyoto University, Japan
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Summary

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order 2 congruence of lines in P3 whose focal surface is a quartic surface X20-d with 20-d ordinary double points. We also show that X15 can be realized as a hyperplane section of the Castelnuovo– Richmond–Igusa quartic hypersurface in P4. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of X15: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in P4 and nonsingular surfaces of degree 10 in P6. Finally we study some birational involutions of a 15- nodal quartic surface which, as it is shown in Part II of the paper jointly with I. Shimada [DS20], belong to a finite set of generators of the group of birational automorphisms of a general 15 nodal quartic surface.

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Recent Developments in Algebraic Geometry
To Miles Reid for his 70th Birthday
 , pp. 66 - 115
Publisher: Cambridge University Press
Print publication year: 2022

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References

Arrondo and Mark Gross. On smooth surfaces in Gr(1, Enrique, P3) with a fundamental curve. Manuscripta Math., 79(3–4):283–298, 1993.CrossRefGoogle Scholar
Avilov, A.. Biregular and birational geometry of double coverings of a projective space with ramification in a quartic with 15 ordinary double points. Izv. Ross. Akad. Nauk Ser. Mat., 83(3):5–14, 2019.CrossRefGoogle Scholar
F. Baker. Principles of geometry. Volume 4. Higher geometry. Cambridge Library Collection. Cambridge University Press, H., Cambridge, 2010. Reprint of the 1925 original.Google Scholar
Cheltsov, Ivan, Alexander Kuznetsov, and Konstantin Shramov. Coble fourfold, öß-invariant quartic threefolds, and Wiman-Edge sextics. Algebra Number Theory, 14(1):213–274, 2020.CrossRefGoogle Scholar
B. Coble. Algebraic geometry and theta functions, volume 10 of American Mathematical Society Colloquium Publications. American Mathematical Society, Arthur, Providence, R.I., 1982. Reprint of the 1929 edition.Google Scholar
Cossec, Francois R.. Reye congruences. Trans. Amer. Math. Soc., 280(2):737–751, 1983.Google Scholar
Dolgachev, Igor, Benson Farb, and Eduard Looijenga. Geometry of the Wiman-Edge pencil, I: algebro-geometric aspects. Eur. J. Math., 4(3):879–930, 2018.Google Scholar
Dolgachev, Igor, and Shigeyuki Kondo. Enriques surfaces, vol. 2. Available online: http://www.math.lsa.umich.edu/^idolga/lecture notes.htmlGoogle Scholar
Dolgachev, Igor and Jonghae Keum. Birational automorphisms of quartic Hessian surfaces. Trans. Amer. Math. Soc., 354(8):3031–3057, 2002.CrossRefGoogle Scholar
V. Dolgachev. Classical algebraic geometry: A modern view. Cambridge University Press, Igor, Cambridge, 2012.CrossRefGoogle Scholar
Dolgachev. Corrado Segre and nodal cubic threefolds. In From classical to modern algebraic geometry, Trends Hist. Sci., Igor, pages 429450. Birkhauser/Springer, Cham, 2016.Google Scholar
Dolgachev, Igor and Ichiro Shimada. 15-nodal quartic surfaces. Part II: the automorphism group. Rend. Circ. Mat. Palermo (2), 69(3):1165–1191, 2020.Google Scholar
L. Edge. A plane sextic and its five cusps. Proc. Roy. Soc. Edinburgh Sect. A, W., 118(3–4):209–223, 1991.CrossRefGoogle Scholar
Endraß, Stephan. On the divisor class group of double solids. Manuscripta Math., 99(3):341–358, 1999.CrossRefGoogle Scholar
Finkelnberg, Hans. Small resolutions of the Segre cubic. Nederl. Akad. Wetensch. Indag. Math., 49(3):261–277, 1987.CrossRefGoogle Scholar
Green, Mark, and Robert Lazarsfeld. Special divisors on curves on a K3 surface. Invent. Math., 89(2):357–370, 1987.CrossRefGoogle Scholar
W. H. T. Hudson. Kummer's quartic surface. Cambridge Mathematical Library. Cambridge University Press, R., Cambridge, 1990. With a foreword by W. Barth, Revised reprint of the 1905 original.Google Scholar
Iliev, Atanas. The Fano surface of the Gushel threefold. Compositio Math., 94(1):81–107, 1994.Google Scholar
Iliev, Atanas, and Laurent Manivel. Fano manifolds of degree ten and EPW sextics. Ann. Sci. Éc. Norm. Supér. (4), 44(3):393–426, 2011.Google Scholar
Iskovskih, V. A.. Fano threefolds. I. Izv. Akad. Nauk SSSR Ser. Mat., 41(3):516–562, 717, 1977.Google Scholar
M. Jessop. A treatise on the line complex. Chelsea Publishing Co., C., New York, 1969.Google Scholar
Johnsen and Andreas Leopold Knutsen. K3 projective models in scrolls, volume 1842 of Lecture Notes in Mathematics. SpringerVerlag, Trygve, Berlin, 2004.CrossRefGoogle Scholar
Kondo, Shigeyuki. The automorphism group of a generic Jacobian Kummer surface. J. Algebraic Geom., 7(3):589–609, 1998.Google Scholar
Kapustka, Grzegorz, and Alessandro Verra. On Morin configurations of higher length. Int. Math. Res. Not., no. 1: 727–772, 2022.Google Scholar
Mukai, Shigeru. Curves, K3 surfaces and Fano 3-folds of genus < 10. In Algebraic geometry and commutative algebra, Vol. I, pages 357377. Kinokuniya, Tokyo, 1988.CrossRefGoogle Scholar
Mukai, Shigeru. Curves and Grassmannians. In Algebraic geometry and related topics (Inchon, 1992), Conf. Proc. Lecture Notes Algebraic Geom., I, pages 19–40. Int. Press, Cambridge, MA, 1993.Google Scholar
Nikulin, V. V.. Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat., 43(1):111–177, 238, 1979.Google Scholar
O'Grady, K. G.. Involutions and linear systems on holomorphic symplectic manifolds. Geom. Funct. Anal., 15(6):1223–1274, 2005.Google Scholar
Kieran, G. O'Grady. Dual double EPW-sextics and their periods. Pure Appl. Math. Q., (Special Issue: In honor of Fedor Bogomolov. Part 1) 4(2):427–468, 2008.Google Scholar
Ran, Ziv. Surfaces of order 1 in Grassmannians. J. Reine Angew. Math., 368:119–126, 1986.CrossRefGoogle Scholar
Rohn, Karl. Die Flachen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung. Math. Ann., 29(1):81–96, 1887.CrossRefGoogle Scholar
Salmon. A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. 7th ed. Vol. 1. Edited by C. H. Rowe. Chelsea Publishing Company, George, New York, 1958.Google Scholar
G. Semple and L. Roth. Introduction to algebraic geometry. Oxford Science Publications. The Clarendon Press, J., Oxford University Press, New York, 1985. Reprint of the 1949 original.Google Scholar
Sturm, R.. Die Gebilde ersten und zweiten Grades der Liniengeometrie in synthetischer Behandlung. vol. 1–3. Leipzig, 1892.Google Scholar
Varley. Weddle's surfaces, Robert, Humbert's curves, and a certain 4-dimensional abelian variety. Amer. J. Math., 108(4):931–951, 1986.CrossRefGoogle Scholar
Verra, Alessandro. Smooth surfaces of degree 9 in G(1,3). Manuscripta Math., 62(4):417–435, 1988.Google Scholar
Vinberg, E. B.. The two most algebraic K3 surfaces. Math. Ann., 265(1):1–21, 1983.CrossRefGoogle Scholar

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