×

zbMATH — the first resource for mathematics

Isomorphic quartic \(K3\) surfaces in the view of Cremona and projective transformations. (English) Zbl 1391.14076
The aim is to study gaps between (1) Cremona equivalence and isomorphism as abstract varieties and (2) Cremona isomorphism and projectively equivalence, for complex smooth quartic surfaces in the projective space.
It is found in Theorem 1.4 that there exists a pair \((S_1,\, S_2)\) of \(K3\) surfaces that is isomorphic as abstract varieties, but not Cremona equivalent, which is shown by an indirect proof. Explicitely, let \(S\) be a \(K3\) surfacae with Néron-Severi lattice being \(\langle h_1,\, h_2\rangle_{\mathbb{Z}}\) with \((h_i^2)_S = 4\) for \(i=1,2\), and \((h_1.h_2)_S=4l\), where \(l\geq 5\), and then \(S_i:=\Phi_{|h_i|}(S)\) for \(i=1,2\) would do.
They also construct a pair \((S_1,\, S_2)\) of \(K3\) surfaces in Theorem 1.5, which are Cremona isomorphic but not projective equivalent. In fact, that a surface \(S\) is a \(K3\) surface with the Néron-Severi lattice \(\langle h_1,\, h_2\rangle_{\mathbb{Z}}\) with \((h_i^2)_S=4\) for \(i=1,2\), and \((h_1.h_2)_S=6\) is equivalent to that the surface \(S\) is the intersection of four hypersurfaces of type \((1,1)\) in \(\mathbb{P}^3\times\mathbb{P}^3\). And the pair \((S_1,\, S_2)\) with \(S_i:=\Phi_{|h_i|}\) for \(i=1,2\) satisfies the desired condition. The key observations are firstly, that the surfaces \(S_1\) and \(S_2\) are determinantal surfaces in the sense of Beauville, secondly to study how automorphisms of \(S\) act on hyperplane sections \(H_1\) and \(H_2\), and on \(S_1\) and \(S_2\), and lastly, to study how automorphisms of \(S\) act on the nowhere-vanishing holomorphic form \(\sigma_S\) of \(S\).
In Appendix, they give a proof for Proposition 1.7 : the existence of certain curves on smooth quartic surfaces in \(\mathbb{P}^3\) when it possesses \(\phi\in\text{Bir}(\mathbb{P}^3)\backslash\text{Aut}(\mathbb{P}^3)\).

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
14E07 Birational automorphisms, Cremona group and generalizations
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid
References:
[1] W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact Complex Surfaces, Second edition, Springer-Verlag, Berlin, 2004. · Zbl 1036.14016
[2] A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), no. 1, 39-64. · Zbl 1076.14534
[3] M. Bhargava, W. Ho and A. Kumar, Orbit parametrizations for K3 surfaces, Forum Math. Sigma 4 (2016), e18, 86 pp. · Zbl 1346.14099
[4] F. Catanese and F. Tonoli, A remarkable moduli space of rank \(6\) vector bundles related to cubic surfaces, in Vector Bundles and Low Codimensional Subvarieties: state of the art and recent developments, 41-87, Quad. Mat. 21, Dept. Math., Seconda Univ. Napoli, Caserta, 2007.
[5] A. Cayley, A memoir on quartic surfaces, Proc. London Math. Soc. S1-3 (1869), no. 1, 19-69. · JFM 03.0390.01
[6] I. V. Dolgachev, Mirror symmetry for lattice polarized \(K3\) surfaces, J. Math. Sci. 81 (1996), no. 3, 2599-2630. · Zbl 0890.14024
[7] D. Festi, A. Garbagnati, B. van Geemen and R. van Luijk, The Cayley-Oguiso automorphism of positive entropy on a K3 surface, J. Mod. Dyn. 7 (2013), no. 1, 75-97. · Zbl 1314.14070
[8] J. Kollár, Private e-mail communications, February 18, 19, 2016.
[9] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 1963/1964, 347-361. · Zbl 0141.37401
[10] M. Mella and E. Polastri, Equivalent birational embeddings, Bull. Lond. Math. Soc. 41 (2009), no. 1, 89-93. · Zbl 1184.14021
[11] —-, Equivalent birational embeddings II: divisors, Math. Z. 270 (2012), no. 3-4, 1141-1161. · Zbl 1259.14011
[12] D. R. Morrison, On \(K3\) surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105-121. · Zbl 0509.14034
[13] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111-177. · Zbl 0408.10011
[14] K. Oguiso, Quartic K3 surfaces and Cremona transformations, in Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, 455-460, Fields Inst. Commun. 67, Springer, New York, 2013. · Zbl 1302.14032
[15] —-, Automorphism groups of Calabi-Yau manifolds of Picard number \(2\), J. Algebraic Geom. 23 (2014), no. 4, 775-795. · Zbl 1304.14051
[16] —-, Smooth quartic K3 surfaces and Cremona transformations, II, (unpublished). arXiv:1206.5049
[17] B. Saint-Donat, Projective models of \(K-3\) surfaces, Amer. J. Math. 96 (1974), no. 4, 602-639. · Zbl 0301.14011
[18] I. Shimada, Private e-mail communications, April 11, 12, 14, 2016.
[19] I. Shimada and T. Shioda, On a smooth quartic surface containing \(56\) lines which is isomorphic as a \(K3\) surface to the Fermat quartic, to appear in Manuscripta Math., 19 pp. · Zbl 1387.14106
[20] N. Takahashi, An application of Noether-Fano inequalities, Math. Z. 228 (1998), no. 1, 1-9. · Zbl 0939.14024
[21] T. T. Truong, Private e-mail communications, February 13, 14, 2016.
[22] K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Mathematics 439, Springer-Verlag, Berlin, 1975. · Zbl 0299.14007
[23] C. Voisin, Hodge Theory and Complex Algebraic Geometry. II, Cambridge Studies in Advanced Mathemathics 77, Cambridge University Press, Cambridge, 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.