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Journal articles on the topic 'Fano fourfolds of K3 type'

Author: Grafiati

Published: 5 October 2024

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1

Fu, Lie, Robert Laterveer, and Charles Vial. "Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 5 (June 1, 2021): 2085–126. http://dx.doi.org/10.1007/s10231-021-01070-0.

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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.

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2

Laterveer, Robert. "On the Chow ring of certain Fano fourfolds." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19, no. 1 (December 1, 2020): 39–52. http://dx.doi.org/10.2478/aupcsm-2020-0004.

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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.

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3

Mongardi, Giovanni. "On symplectic automorphisms of hyper-Kähler fourfolds of K3[2] type." Michigan Mathematical Journal 62, no. 3 (September 2013): 537–50. http://dx.doi.org/10.1307/mmj/1378757887.

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4

Laza, Radu, and Kieran O’Grady. "Birational geometry of the moduli space of quartic surfaces." Compositio Mathematica 155, no. 9 (August 2, 2019): 1655–710. http://dx.doi.org/10.1112/s0010437x19007516.

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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.

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5

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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6

Pym, Brent. "Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets." Compositio Mathematica 153, no. 4 (March 13, 2017): 717–44. http://dx.doi.org/10.1112/s0010437x16008174.

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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.

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7

Konovalov, V. A. "THE USE OF MARKOV ALGORITHMS FOR THE STUDY OF l-VOIDS IN BIG DATA OF SOCIO-ECONOMIC SYSTEMS. PART 2." Vestnik komp'iuternykh i informatsionnykh tekhnologii, no. 217 (July 2022): 30–41. http://dx.doi.org/10.14489/vkit.2022.07.pp.030-041.

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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.

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8

Kretschmer, Andreas. "The Chow ring of hyperkähler varieties of $$K3^{[2]}$$-type via Lefschetz actions." Mathematische Zeitschrift, September 9, 2021. http://dx.doi.org/10.1007/s00209-021-02846-z.

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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.

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9

Huybrechts, Daniel. "Chow groups of surfaces of lines in cubic fourfolds." Épijournal de Géométrie Algébrique Special volume in honour of... (July 30, 2023). http://dx.doi.org/10.46298/epiga.2023.10425.

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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.

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10

Gounelas, Frank, and Alexis Kouvidakis. "On some invariants of cubic fourfolds." European Journal of Mathematics 9, no. 3 (July 11, 2023). http://dx.doi.org/10.1007/s40879-023-00651-y.

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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.

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11

Oberdieck, Georg. "Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.10.

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Abstract We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$ -type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds.

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12

Frei, Sarah, and Katrina Honigs. "Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci." Forum of Mathematics, Sigma 11 (2023). http://dx.doi.org/10.1017/fms.2023.37.

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Abstract We describe the Galois action on the middle $\ell $ -adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v. We show this action is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat {A})[3]$ , which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel over ${\mathbb C}$ [21]. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$ . Over ${\mathbb C}$ , they are known to be symplectic and contained in the kernel of the map $\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$ . We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$ . When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ where A is $(1,3)$ -polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$ .

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13

Bangere, Purnaprajna, Francisco Javier Gallego, and Miguel González. "Deformations of Hyperelliptic and Generalized Hyperelliptic Polarized Varieties." Mediterranean Journal of Mathematics 20, no. 2 (January 29, 2023). http://dx.doi.org/10.1007/s00009-023-02278-5.

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AbstractThe purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let $$\varphi : X \longrightarrow Y \subset \textbf{P}^N$$ φ : X ⟶ Y ⊂ P N be the morphism induced by |L|; when does $$\varphi $$ φ deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension $$m \ge 2$$ m ≥ 2 , sectional genus g and $$L^m=2g$$ L m = 2 g . This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the $$L^m=2g$$ L m = 2 g case, with one exception, a general deformation of $$\varphi $$ φ is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, $$\varphi $$ φ rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus $$g \ge 3$$ g ≥ 3 . Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.

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14

Nesterov, Denis, and Georg Oberdieck. "Elliptic Curves in Hyper-Kähler Varieties." International Mathematics Research Notices, February 14, 2020. http://dx.doi.org/10.1093/imrn/rnaa016.

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Abstract We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3,780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed $j$-invariant in terms of Gromov–Witten invariants. In $K3^{[2]}$-type this leads to explicit formulas of these counts in terms of modular forms.

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15

Fatighenti, Enrico. "Examples of Non-Rigid, Modular Vector Bundles on Hyperkähler Manifolds." International Mathematics Research Notices, February 19, 2024. http://dx.doi.org/10.1093/imrn/rnae021.

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Abstract We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3$^{[2]}$-type, which move in a 20-dimensional family and study their algebraic properties. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic four-fold and the Debarre–Voisin hyperkähler manifold.

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16

Ascher, Kenneth, Kristin DeVleming, and Yuchen Liu. "Wall crossing for K‐moduli spaces of plane curves." Proceedings of the London Mathematical Society 128, no. 6 (June 2024). http://dx.doi.org/10.1112/plms.12615.

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AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

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