Browsing by Subject "K3-surface"
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Item Open Access 800 conics on a smooth quartic surface(Elsevier BV * North-Holland, 2022-03-10) Degtyarev, AlexWe construct an example of a smooth spatial quartic surface that contains 800 irreducible conics. © 2022 Elsevier B.V.Item Open Access Conics on Barth-Bauer octics(Zhongguo Kexue Zazhishe / Science in China Press, 2024-04-23) Degtyarev, AlexWe analyze the configurations of conics and lines on a special class of Kummer octic surfaces. In particular, we bound the number of conics by 176 and show that there is a unique surface with 176 conics, all irreducible: it admits a faithful action of one of the Mukai groups. Therefore, we also discuss conics and lines on Mukai surfaces: we discover a double plane (ramified at a smooth sextic curve) that contains 8,910 smooth conics.Item Open Access Deformation classes of singular quartic surfaces(2016-12) Aktaş, Çisem GüneşWe study complex spatial quartic surfaces with simple singularities and give their classication up to equisingular deformation. Simple quartics are K3-surfaces and as such they can be studied by means of the global Torelli theorem and the surjectivity of the period map combined with Nikulin's theory of discriminant forms. We reduce the classification problem to a certain arithmetical problem concerning lattice extensions. Then, based on Nikulin's existence criterion, we list all sets of simple singularities realized by non-special quartics; the result is stated in terms of perturbations of a few extremal sets. For each realizable set of singularities, we use Miranda{Morrison's theory to give a complete description of the connected components of the corresponding equisingular stratum.Item Open Access Lines in supersingular quartics(Nihon Sugakkai,Mathematical Society of Japan, 2021-10-19) Degtyarev, AlexWe show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.Item Open Access Lines on K3-quartics via triangular sets(Springer, 2024-10-06) Degtyarev, Alex; Rams, SlawomirWe prove the sharp upper bound of at most 52 lines on a complex K3-surface of degree 4 with a non-empty singular locus. We also classify the configurations of more than 48 lines on smooth complex quartics.Item Open Access Lines on smooth polarized K3-surfaces(Springer, 2019) Degtyarev, AlexFor each integer D⩾3D⩾3, we give a sharp bound on the number of lines contained in a smooth complex 2D-polarized K3-surface in PD+1PD+1. In the two most interesting cases of sextics in P4P4 and octics in P5P5, the bounds are 42 and 36, respectively, as conjectured in an earlier paper.Item Open Access Smooth models of singular K3-surfaces(European Mathematical Society Publishing House, 2019) Degtyarev, AlexanderWe 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.Item Open Access Tritangents to smooth sextic curves(Association des Annales de l'Institut Fourier, 2022-10-21) Degtyarev, AlexWe prove that a smooth plane sextic curve can have at most 72 tritangents, whereas a smooth real sextic may have at most 66 real tritangents. © 2022 Association des Annales de l'Institut Fourier. All rights reserved.