Zariski-van Kampen Method and Transcendental Lattices of Certain Singular $K3$ Surfaces

Zariski-van Kampen Method and Transcendental Lattices of Certain Singular $K3$ Surfaces

Ken-ichiro ARIMA and Ichiro SHIMADA

Source: Tokyo J. of Math. Volume 32, Number 1 (2009), 201-227.

Abstract

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular $K3$ surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over $\mathbf{Q}(\sqrt{5})$ and are conjugate to each other by the action of $\text{Gal}(\mathbf{Q}(\sqrt{5})/\mathbf{Q})$.

Primary Subjects: 14J28

Secondary Subjects: 14H50, 14H25

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Permanent link to this document: http://projecteuclid.org/euclid.tjm/1249648417
Digital Object Identifier: doi:10.3836/tjm/1249648417
Zentralblatt MATH identifier: 05604243
Mathematical Reviews number (MathSciNet): MR2541164

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References

E. Artal-Bartolo, Sur les couples de Zariski. J. Algebraic Geom., 3 (1994), 223--247.

Mathematical Reviews (MathSciNet): MR1257321

Zentralblatt MATH: 0823.14013

E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agust\' \i n, On sextic curves with big Milnor number, In, Trends in singularities, Trends Math., 1--29. Birkhäuser, Basel, 2002.

Mathematical Reviews (MathSciNet): MR1900779

Zentralblatt MATH: 1018.14009

E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustí n, Effective invariants of braid monodromy, Trans. Amer. Math. Soc., 359 (2007) 165--183 (electronic).

Mathematical Reviews (MathSciNet): MR2247887

Zentralblatt MATH: 1109.14013

Digital Object Identifier: doi:10.1090/S0002-9947-06-03881-5

J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, volume 290 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, third edition,, 1999.

Mathematical Reviews (MathSciNet): MR1662447

A. Degtyarev, On deformations of singular plane sextics, J. Algebraic Geom., 17 (2008), 101--135.

Mathematical Reviews (MathSciNet): MR2357681

Zentralblatt MATH: 1131.14040

Carl Friedrich Gauss, Disquisitiones arithmeticae, Springer-Verlag, New York, 1986. Translated and with a preface by Arthur A. Clarke, Revised by William C. Waterhouse, Cornelius Greither and A. W. Grootendorst and with a preface by Waterhouse.

Mathematical Reviews (MathSciNet): MR837656

K. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology, 20 (1981), 15--51.

Mathematical Reviews (MathSciNet): MR592569

Digital Object Identifier: doi:10.1016/0040-9383(81)90013-6

V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), 111--177, 238. English translation: Math USSR-Izv. 14 (1979), no. 1, 103--167 (1980).

Mathematical Reviews (MathSciNet): MR525944

M. Oka, Symmetric plane curves with nodes and cusps, J. Math. Soc. Japan, 44 (1992), 375--414.

Mathematical Reviews (MathSciNet): MR1167373

Zentralblatt MATH: 0767.14011

Digital Object Identifier: doi:10.2969/jmsj/04430375

Project Euclid: euclid.jmsj/1227107463

U. Persson, Horikawa surfaces with maximal Picard numbers, Math. Ann., 259 (1982), 287--312.

Mathematical Reviews (MathSciNet): MR661198

Zentralblatt MATH: 0466.14010

Digital Object Identifier: doi:10.1007/BF01456942

U. Persson, Double sextics and singular $K$-$3$ surfaces, in, Algebraic geometry, Sitges (Barcelona), 1983, volume 1124 of Lecture Notes in Math., 262--328. Springer, Berlin, 1985.

Mathematical Reviews (MathSciNet): MR805337

Zentralblatt MATH: 0603.14023

Digital Object Identifier: doi:10.1007/BFb0075003

I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli's theorem for algebraic surfaces of type $K3$, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 530--572, Reprinted in I. R. Shafarevich, Collected Mathematical Papers, Springer-Verlag, Berlin, 1989, 516--557.

Mathematical Reviews (MathSciNet): MR284440

M. Roczen, Recognition of simple singularities in positive characteristic, Math. Z., 210 (1992), 641--653.

Mathematical Reviews (MathSciNet): MR1175728

Zentralblatt MATH: 0774.14002

Digital Object Identifier: doi:10.1007/BF02571820

Matthias Schütt, Fields of definition of singular $K3$ surfaces, Commun. Number Theory Phys., 1 (2007), 307--321.

Mathematical Reviews (MathSciNet): MR2346573

Zentralblatt MATH: 1157.14308

I. Shimada, Fundamental groups of open algebraic varieties, Topology, 34 (1995), 509--531.

Mathematical Reviews (MathSciNet): MR1341806

Digital Object Identifier: doi:10.1016/0040-9383(94)00045-M

I. Shimada, A note on Zariski pairs, Compositio Math., 104 (1996), 125--133.

Mathematical Reviews (MathSciNet): MR1421396

Zentralblatt MATH: 0878.14018

I. Shimada, Picard-Lefschetz theory for the universal coverings of complements to affine hypersurfaces, Publ. Res. Inst. Math. Sci., 32 (1996), 835--927.

Mathematical Reviews (MathSciNet): MR1433320

Digital Object Identifier: doi:10.2977/prims/1195162384

Project Euclid: euclid.prims/1195162384

I. Shimada, On the Zariski-van Kampen theorem. Canad. J. Math., 55 (2003), 133--156.

Mathematical Reviews (MathSciNet): MR1952329

I. Shimada, On arithmetic Zariski pairs in degree 6, Adv. Geom., 8(2) (2008), 205--225.

Mathematical Reviews (MathSciNet): MR2405237

Zentralblatt MATH: 1153.14017

Digital Object Identifier: doi:10.1515/ADVGEOM.2008.015

I. Shimada, Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface, Trans. Amer. Math. Soc., 361 (2) (2009), 909--949.

Mathematical Reviews (MathSciNet): MR2452829

Zentralblatt MATH: 05518627

Digital Object Identifier: doi:10.1090/S0002-9947-08-04560-1

I. Shimada, Non-homeomorphic conjugate complex varieties, 2007. preprint, arXiv:math/0701115.

T. Shioda and H. Inose, On singular $K3$ surfaces, in, Complex analysis and algebraic geometry, 119--136. Iwanami Shoten, Tokyo, 1977.

Mathematical Reviews (MathSciNet): MR441982

Zentralblatt MATH: 0374.14006

T. Shioda, $K3$ surfaces and sphere packings, J. Math. Soc. Japan, 60(4) (2008), 1083--1105.

Mathematical Reviews (MathSciNet): MR2467871

Zentralblatt MATH: 05500752

Digital Object Identifier: doi:10.2969/jmsj/06041083

Project Euclid: euclid.jmsj/1225894034

Jin-Gen Yang, Sextic curves with simple singularities, Tohoku Math. J., 48 (1996), 203--227.

Mathematical Reviews (MathSciNet): MR1387816

Digital Object Identifier: doi:10.2748/tmj/1178225377

Project Euclid: euclid.tmj/1178225377

O. Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math., 51 (1929), 305--328.

Mathematical Reviews (MathSciNet): MR1506719

Digital Object Identifier: doi:10.2307/2370712

JSTOR: links.jstor.org

O. Zariski, The Topological Discriminant Group of a Riemann Surface of Genus $p$, Amer. J. Math., 59 (1937), 335--358.

Mathematical Reviews (MathSciNet): MR1507244

Digital Object Identifier: doi:10.2307/2371416

JSTOR: links.jstor.org

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