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Journal articles on the topic 'Special cubic fourfolds'

Author: Grafiati

Published: 5 October 2024

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1

Addington, Nicolas, and Asher Auel. "Some Non-Special Cubic Fourfolds." Documenta Mathematica 23 (2018): 637–51. http://dx.doi.org/10.4171/dm/628.

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2

Truong, Hoang Le, and Hoang Ngoc Yen. "A note on special cubic fourfolds of small discriminants." Forum Mathematicum 33, no. 5 (August 26, 2021): 1137–55. http://dx.doi.org/10.1515/forum-2020-0355.

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Abstract In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfolds X ∈ 𝒞 δ {X\in\mathcal{C}_{\delta}} for 6 < δ ≤ 30 {6<\delta\leq 30} and δ ≡ 0 ( mod 6 ) {\delta\equiv 0~{}(\bmod~{}6)} . This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants.

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3

Li, Zhiyuan, and Letao Zhang. "Modular forms and special cubic fourfolds." Advances in Mathematics 245 (October 2013): 315–26. http://dx.doi.org/10.1016/j.aim.2013.06.003.

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4

Tanimoto, Sho, and Anthony Várilly-Alvarado. "Kodaira dimension of moduli of special cubic fourfolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 752 (July 1, 2019): 265–300. http://dx.doi.org/10.1515/crelle-2016-0053.

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Abstract A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors {{\mathcal{C}}_{d}} in the moduli space {{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of {{\mathcal{C}}_{d}} . For example, if {d=6n+2} , then we show that {{\mathcal{C}}_{d}} is of general type for {n>18} , {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if {n>13} and {n\neq 15} . In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

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5

Laterveer, Robert. "Algebraic cycles and very special cubic fourfolds." Indagationes Mathematicae 30, no. 2 (March 2019): 317–28. http://dx.doi.org/10.1016/j.indag.2018.12.002.

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6

Pertusi, Laura. "Fourier–Mukai partners for very general special cubic fourfolds." Mathematical Research Letters 28, no. 1 (2021): 213–43. http://dx.doi.org/10.4310/mrl.2021.v28.n1.a9.

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7

Bülles, Tim-Henrik. "Motives of moduli spaces on K3 surfaces and of special cubic fourfolds." manuscripta mathematica 161, no. 1-2 (November 7, 2018): 109–24. http://dx.doi.org/10.1007/s00229-018-1086-0.

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8

Kuznetsov, Alexander, and Alexander Perry. "Derived categories of Gushel–Mukai varieties." Compositio Mathematica 154, no. 7 (May 25, 2018): 1362–406. http://dx.doi.org/10.1112/s0010437x18007091.

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We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

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9

Bayer, Arend, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, and Paolo Stellari. "Stability conditions in families." Publications mathématiques de l'IHÉS 133, no. 1 (May 17, 2021): 157–325. http://dx.doi.org/10.1007/s10240-021-00124-6.

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AbstractWe develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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10

Pelzl, J., and C. Dimitropoulos. "Effect of Deuteration on the Phase Transitions and on the Critical Dynamics in Ammonium Hexachlorometallates." Zeitschrift für Naturforschung A 49, no. 1-2 (February 1, 1994): 232–46. http://dx.doi.org/10.1515/zna-1994-1-235.

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Abstract Recent and novel data obtained from chlorine NQR measurements on natural and deuterated (NH4)2MCl6 compounds are discussed with special regard to the influence of the ammonium-ion dynamics on the structural stability of these crystals. The temperature dependence (4.2 K to 350 K) of the chlorine NOR frequency vQ and relaxation rates T1-1 , T2-1 obtained from the natural ammonium salts of Sn, Pd, Os, Pb, Te, Se and from the deuterated salts of Sn, Te and Se are analysed. Slight deviations from the normal temperature behaviour of vQ and T1-1 are found in Sn, Pd and Os compounds which stay cubic in the whole temperature range. The ammonium compounds of Pb and Te undergo a structural transformation between 80 K and 90 K from the cubic to a trigonal phase which is distinguished by the preservation of the single line spectrum of the chlorine NQR below rel. The observed divergence of T1-1 at the transition point can be described in terms of a spin-phonon process in the presence of an overdamped soft mode. Deuteration of (NH4)2TeCl6 only slightly affects the transition of Tc2 but leads to new structural changes at lower temperatures. Whereas the natural compound stays trigonal down to 4.2 K the deuterated crystal undergoes two additional structural transformations at Tc2 = 48 K and Tc3 = 28 K which are correlated with a slowing down of the deuteron motion. Approaching Tc2 from above, the spin-lattice relaxation rate and the spin-spin relaxation rate of the chlorine NQR exhibit distinct anomalies which are attributed to limited jumps of the octahedron in a shallow potential. The barrier height of this potential deduced from the chlorine NQR spin-lattice relaxation rate is 400 K. The transition at Tc2 is explained by the condensation in one minimum of this potential. At Tc3 a long range correlation is formed which is accompanied by a rotation of the octahedron about its fourfold axis. A similar mechanism is adopted for the transitions observed in (NH4)2SeCl6 at Tc= 24 K and in (ND4)2 SeCl6 at Tc = 48 K.

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11

Kim, Yeongrak. "Ulrich bundles on special cubic fourfolds of small discriminants." Journal of Algebra, July 2024. http://dx.doi.org/10.1016/j.jalgebra.2024.07.028.

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12

Nuer, Howard. "Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces." Algebraic Geometry, May 1, 2017, 281–89. http://dx.doi.org/10.14231/ag-2017-015.

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13

BRAKKEE, EMMA. "Two polarised K3 surfaces associated to the same cubic fourfold." Mathematical Proceedings of the Cambridge Philosophical Society, March 16, 2020, 1–14. http://dx.doi.org/10.1017/s0305004120000055.

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Abstract For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilb n (S) and Hilb n (Sτ) are birational.

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14

Pertusi, Laura, and Paolo Stellari. "Categorical Torelli theorems: results and open problems." Rendiconti del Circolo Matematico di Palermo Series 2, September 15, 2022. http://dx.doi.org/10.1007/s12215-022-00796-x.

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AbstractWe survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.

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15

Bricalli, Davide, and Filippo Francesco Favale. "Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras." Collectanea Mathematica, November 19, 2022. http://dx.doi.org/10.1007/s13348-022-00382-5.

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AbstractIn this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.

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16

Colliot-Thélène, Jean-Louis, and Alena Pirutka. "Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable." Épijournal de Géométrie Algébrique Volume 2 (December 10, 2018). http://dx.doi.org/10.46298/epiga.2018.volume2.3950.

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En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de Hodge enti\`ere pour les classes de degr\'e 4 vaut pour les vari\'et\'es projectives et lisses de dimension 4 fibr\'ees en solides cubiques au-dessus d'une courbe, sans restriction sur les fibres singuli\`eres. --------------- We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension $2$ Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree $4$ classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres. Comment: in French

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