Littérature scientifique sur le sujet « Special cubic fourfolds »

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.

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.

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.

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.

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.

Dans cette thèse, nous étudions les transformations birationnelles spéciales des quadriques lisses. Nous obtenons un résultat de classification en dimensions 3 et 4. Dans ces deux cas, nous démontrons qu'il n'existe qu'un seul exemple. Pour la dimension 3, il est défini par le système linéaire de quadriques passant par une courbe rationnelle normale quartique. Pour la dimension 4, il est défini par le système linéaire de cubiques passant par une surface K3 non minimale de degré 10 avec 2 (-1)-droites disjointes qui n'est contenue dans aucune autre quadrique. Le lieu de base de la transformation inverse est en général une surface lisse du même type. De plus, nous montrons que les surfaces K3 correspondantes sont des partenaires de Fourier-Mukai non isomorphes. Ces surfaces sont également liées aux cubiques de dimension 4 spéciales. Plus précisément, nous montrons qu'une cubique générale dans le diviseur de Hassett des cubiques spéciales de discriminante 14 contient une telle surface. Il s'agit du premier exemple d'une famille de surfaces non rationnelles caractérisant les cubiques dans ce diviseur. L'étude des transformations birationnelles spéciales des quadriques est motivée par un exemple décrit par M. Bernardara, E. Fatighenti, L. Manivel, et F. Tanturri, qui ont fourni une liste de 64 nouvelles familles de variétés de Fano de type K3. De nombreux exemples dans leur liste donnent des variétés qui admettent des contractions birationnelles multiples, réalisées comme des éclatements des variétés de Fano le long des surfaces K3 non minimales. La nature des constructions implique que les surfaces K3 ont des catégories dérivées équivalentes. Nous répondons partiellement à la question naturelle : Pour quelles familles les surfaces K3 correspondantes sont-elles isomorphes, et pour quelles familles ne le sont-elles pas ?
In this thesis we study special self-birational transformations of smooth quadrics. We obtain a classification result in dimensions 3 and 4. In these two cases, we prove that there is only one example. In the case of dimension 3, it is given by the linear system of quadrics passing through a rational normal quartic curve. In the case of dimension 4, it is given by the linear system of cubic complexes passing through a non-minimal K3 surface of degree 10 with 2 skew (-1)-lines that is not contained in any other quadric. The base locus scheme of the inverse map is in general a smooth surface of the same type. Moreover, we prove that the corresponding pair of K3 surfaces are non-isomorphic Fourier-Mukai parters. These surfaces are also related to special cubic fourfolds. More precisely, we show that a general cubic in the Hassett divisor of special cubic fourfolds of discriminant 14 contains such a surface. This is the first example of a family of non-rational surfaces characterizing cubics in this divisor. The study of special birational transformations of quadrics is motivated by an example described by M. Bernardara, E. Fatighenti, L. Manivel, et F. Tanturri, who provided a list of 64 new families of Fano fourfolds of K3 type. Many examples in their list give varieties that admit multiple birational contractions realized as blow-ups of Fano manifolds along non-minimal K3 surfaces. The nature of the constructions implies that the corresponding K3 surfaces have equivalent derived categories. We partially answer the natural question: for which families the corresponding K3 surfaces are isomorphic, and for which families they are not?