On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

Shimada, Ichio; Shioda, Tetsuji

doi:10.1007/s00229-016-0886-3

Cited by 15 publications

(21 citation statements)

References 21 publications

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“…Shioda first noticed that X 48 and X 56 are isomorphic to each other as abstract K3 surfaces. An explicit equation of X 56 and an explicit isomorphism between X 48 and X 56 were found by Shimada and Shioda [16]. The two surfaces are not projectively equivalent to each other, but they form an Oguiso pair.…”

Section: An Explicit Oguiso Pairmentioning

confidence: 91%

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Symmetries and equations of smooth quartic surfaces with many lines

Veniani

2019

Rev. Mat. Iberoam.

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Section: An Explicit Oguiso Pairmentioning

confidence: 91%

“…Starting from this equation, we provide here a way to compute an explicit isomorphism between Q 54 and X ′′ 52 following a method illustrated by Shimada and Shioda. We refer to their article [16] for further details on the algorithms used.…”

Section: 41mentioning

confidence: 99%

Symmetries and equations of smooth quartic surfaces with many lines

Veniani

2019

Rev. Mat. Iberoam.

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“…Finally, the three projective quartics obtained are identified with the classical Fermat quartic X 48 or the pair X 56 ,X 56 constructed in [5] and studied further in [25] according to the number of lines contained in the surface.…”

Section: 3mentioning

confidence: 99%

Smooth models of singular $K3$-surfaces

Degtyarev

2019

Rev. Mat. Iberoam.

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We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of singular K3-surfaces of small discriminant. As a by-product, we observe a correlation (up to a certain limit) between the discriminant of a singular K3-surface and the number of lines in its models. We also construct a K3-quartic surface with 52 lines and singular points, as well as a few other examples with many lines or models. 64 [2, 0, 32] Y 56 (4, 4) 32 (3, 7) 24 64 [4, 0, 16] ?? Conjecturally, none 64 [8, 0, 8] X 56 * (4, 6) 8 (4, 4) 32 (2, 8) 16 (see page 22) X 48 (2, 8) 48 (see page 22); Mukai group F 384 75 [10, 5, 10] Q ′′′ 52 (5, 0) 4 (3, 6) 48 76 [2, 0, 38] Y ′ 52 (4, 6) 2 (4, 4) 16 (3, 5) 20 (2, 8) 14 [8, 2, 10] * Ỹ ′ 48 (3, 5) 24 (2, 8) 24 76 [4, 2, 20] Q 54 (4, 4) 24 (4, 3) 24 (0, 12) 6 ; see (D1) X ′′ 52 (6, 0) 1 (4, 4) 9 (4, 3) 18 (3, 5) 18 (0, 12) 6 ; see (D1) 79 [2, 1, 40] Y ′′ 52 (4, 5) 8 (4, 3) 12 (3, 6) 16 (2, 7) 16 [4, 1, 20] * Ỹ ′′ 48 (4, 3) 6 (3, 6) 12 (2, 7) 30 [8, 1, 10] * 80 [4, 0, 20] Z 52 (6, 0) 4 (4, 4) 12 (4, 2) 24 (2, 8) 12 ; see (D2), (D5) Z 50 * (4, 4) 10 (3, 5) 40 Z ′ 48 (4, 2) 16 (2, 8) 32 ; see (D2) Z ′′ 48 * (3, 5) 48 ; see (D2), (D5) 80 [8, 4, 12] X ′ 52 (

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“…This does not occur for general a k ij , though this needs proof. See [SS16,Ogu16] for this and other such examples.…”

Section: Example 33 Start With Pmentioning

confidence: 99%

Algebraic hypersurfaces

Kollár

2019

Bull. Amer. Math. Soc.

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We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss these questions without any previous knowledge of algebraic geometry. At the end we formulate the main recent results and state the most important open questions. Algebraic geometry started as the study of plane curves C ⊂ R 2 defined by a polynomial equation and later extended to surfaces and higher dimensional sets defined by systems of polynomial equations. Besides using R n , it is frequently more advantageous to work with C n or with the corresponding projective spaces RP n and CP n. Later it was realized that the theory also works if we replace R or C by other fields, for example the field of rational numbers Q or even finite fields F q. When we try to emphasize that the choice of the field is pretty arbitrary, we use A n to denote affine n-space and P n to denote projective n-space.

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On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

Abstract: Abstract. We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines. We give the isomorphism explicitly.

Cited by 15 publications

References 21 publications

Symmetries and equations of smooth quartic surfaces with many lines

Symmetries and equations of smooth quartic surfaces with many lines

Smooth models of singular $K3$-surfaces

Algebraic hypersurfaces

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