Academic literature on the topic 'Fano fourfolds of K3 type'

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

AbstractWe prove that certain Fano fourfolds of K3 type constructed by Fatighenti–Mongardi have a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

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.

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities$\widetilde{E}_{6},\widetilde{E}_{7}$and$\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type$q_{5,1}$are the only log symplectic structures on projective four-space whose singular points are all elliptic.

The second part of the article is presented. Big data l-voids are considered using the N-scheme of the Markov algorithm. The diagrams of occurrences of l-voids in a semi-Eulerian cycle containing an Euler path, a matroid and an incomplete Fano matroid, minors K3, 3 and K5, an extra large cycle of occurrences are analyzed. An example of reconstructing a fragment of an incomplete Fano matroid with l-voids is considered. Examples are given for independent implementation of the method of filling the artificial intelligence database (AnwM f typeK) DB based on the results of the analysis of l-voids in the N-scheme of the Markov algorithm. The database is populated using a formal language with the alphabet M = litj/abdgckm where litj are terminal elements, abdgckm are non-terminal elements, and the basis of the alphabet is i – initial, t – terminal, j – isomorphic, l-empty morphisms. The p – initial, e – simple, and h – final occurrences of words from the Markov alphabet A are determined. An example of a simplified numbering of occurrence chains is given, which ensures the selection of all objects represented by words in the Markov alphabet A, belonging to “their own” chain of occurrences, as well as all related to chain object. The mechanism of dynamic change of the number-type in the N-scheme of the Markov algorithm is presented, which provides the analysis of the word as an evolving category, as well as the evolution of categories, which includes such a word. It has been established that normal Markov inference is characterized by temporal conditionality and locality. The conclusion is made about the expediency of considering the N-scheme in a distributed computing system. In this case, it is necessary to analyze the diagrams of occurrences of objects that have several designations with words from the Markov alphabet A, in order to identify them and process data from such objects.

AbstractWe propose an explicit conjectural lift of the Neron–Severi Lie algebra of a hyperkähler variety X of $$K3^{[2]}$$ K 3 [ 2 ] -type to the Chow ring of correspondences $$\mathrm{CH}^*(X \times X)$$ CH ∗ ( X × X ) in terms of a canonical lift of the Beauville–Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a K3 surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of X of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the cohomological grading operator.

The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.

AbstractFor a general cubic fourfold $$X\subset \mathbb {P}^5$$ X ⊂ P 5 with Fano variety F, we compute the Hodge numbers of the locus $$S\subset F$$ S ⊂ F of lines of second type and the class of the locus $$V\subset F$$ V ⊂ F of triple lines, using the description of the latter in terms of flag varieties. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.

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?