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Автор: Grafiati

Опубліковано: 7 липня 2024

Оновлено: 7 липня 2024

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1

Aprodu, Marian, and Gavril Farkas. "Green’s conjecture for curves on arbitrary K3 surfaces." Compositio Mathematica 147, no. 3 (February 15, 2011): 839–51. http://dx.doi.org/10.1112/s0010437x10005099.

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AbstractGreen’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

2

Bini, Gilberto, Robert Laterveer, and Gianluca Pacienza. "Voisin’s conjecture for zero-cycles on Calabi–Yau varieties and their mirrors." Advances in Geometry 20, no. 1 (January 28, 2020): 91–108. http://dx.doi.org/10.1515/advgeom-2019-0008.

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AbstractWe study a conjecture, due to Voisin, on 0-cycles on varieties with pg = 1. Using Kimura’s finite dimensional motives and recent results of Vial’s on the refined (Chow–)Künneth decomposition, we provide a general criterion for Calabi–Yau manifolds of dimension at most 5 to verify Voisin’s conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to 5.

3

Shen, Junliang, Qizheng Yin, and Xiaolei Zhao. "Derived categories of surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties." Compositio Mathematica 156, no. 1 (November 26, 2019): 179–97. http://dx.doi.org/10.1112/s0010437x19007735.

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Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

4

Schreieder, Stefan. "Refined unramified cohomology of schemes." Compositio Mathematica 159, no. 7 (June 15, 2023): 1466–530. http://dx.doi.org/10.1112/s0010437x23007236.

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We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$ . This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$ . We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$ -adic analogues of these results over any field $k$ which contains all $\ell$ -power roots of unity.

5

Raicu, Claudiu, and Steven V. Sam. "Bi-graded Koszul modules, K3 carpets, and Green's conjecture." Compositio Mathematica 158, no. 1 (January 2022): 33–56. http://dx.doi.org/10.1112/s0010437x21007703.

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We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the canonical ribbon conjecture of Bayer and Eisenbud holds over a field of characteristic $0$ or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.

6

Shen, Junliang, and Qizheng Yin. "CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (November 5, 2018): 1601–27. http://dx.doi.org/10.1017/s147474801800049x.

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We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov’s noncommutative $K3$ category associated to a nonsingular cubic 4-fold.By introducing a filtration on the $\text{CH}_{1}$-group of a cubic 4-fold $Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$ category ${\mathcal{A}}_{Y}$. This is a noncommutative analog of O’Grady’s conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.Our method provides systematic constructions of (a) the Beauville–Voisin filtration on the $\text{CH}_{0}$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in ${\mathcal{A}}_{Y}$.

7

Charles, François, and Alena Pirutka. "La conjecture de Tate entière pour les cubiques de dimension quatre." Compositio Mathematica 151, no. 2 (October 16, 2014): 253–64. http://dx.doi.org/10.1112/s0010437x14007386.

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AbstractWe prove the integral Tate conjecture for cycles of codimension$2$on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from$2$or$3$. The proof relies on the Tate conjecture with rational coefficients, proved in that setting by the first author, and on an argument of Voisin coming from complex geometry.

8

Pirutka, Alena. "Invariants birationnels dans la suite spectrale de Bloch-Ogus." Journal of K-theory 10, no. 3 (June 7, 2012): 565–82. http://dx.doi.org/10.1017/is012004021jkt191.

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AbstractFor a field k of cohomological dimension d we prove that the groups , (l, car.k) = 1, are birational invariants of smooth projective geometrically integral varieties over k of dimension n. Using the Kato conjecture, which has been recently established by Kerz and Saito [18], we obtain a similar result over a finite field for the groups . We relate one of these invariants with the cokernel of the l-adic cycle class map , which gives an analogue of a result of Colliot-Thélène and Voisin [5] 3.11 over ℂ for varieties over a finite field.

9

Laterveer, Robert. "Some Calabi–Yau fourfolds verifying Voisin’s conjecture." Ricerche di Matematica 67, no. 2 (January 15, 2018): 401–11. http://dx.doi.org/10.1007/s11587-018-0352-5.

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10

Martin, Olivier. "On a conjecture of Voisin on the gonality of very general abelian varieties." Advances in Mathematics 369 (August 2020): 107173. http://dx.doi.org/10.1016/j.aim.2020.107173.

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11

Laterveer, Robert. "Some results on a conjecture of Voisin for surfaces of geometric genus one." Bollettino dell'Unione Matematica Italiana 9, no. 4 (March 12, 2016): 435–52. http://dx.doi.org/10.1007/s40574-016-0060-6.

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12

Gao, Ziyang. "Generic rank of Betti map and unlikely intersections." Compositio Mathematica 156, no. 12 (December 2020): 2469–509. http://dx.doi.org/10.1112/s0010437x20007435.

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Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

13

PETERS, CHRIS. "BLOCH-TYPE CONJECTURES AND AN EXAMPLE A THREE-FOLD OF GENERAL TYPE." Communications in Contemporary Mathematics 12, no. 04 (August 2010): 587–605. http://dx.doi.org/10.1142/s0219199710003932.

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The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method [19] (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A three-fold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.

14

Dan, Ananyo. "On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci." Communications in Contemporary Mathematics 17, no. 05 (October 2015): 1550002. http://dx.doi.org/10.1142/s0219199715500029.

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For any integer d ≥ 5, the Noether–Lefschetz locus, denoted NL d, parametrizes smooth degree d surfaces in ℙ3 with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of NL d is of codimension (in the space of all smooth surfaces in ℙ3 of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of NL d, denoted NL r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of NL r,d is of codimension [Formula: see text] and the irreducible component of NL r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d.

15

Laterveer, Robert, and Charles Vial. "On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties." Canadian Journal of Mathematics 72, no. 2 (September 3, 2019): 505–36. http://dx.doi.org/10.4153/s0008414x19000191.

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AbstractThis note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

16

Tian, Zhiyu, and Hong R. Zong. "One-cycles on rationally connected varieties." Compositio Mathematica 150, no. 3 (March 2014): 396–408. http://dx.doi.org/10.1112/s0010437x13007549.

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AbstractWe prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

17

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.

18

Laterveer, Robert. "Zero-cycles on self-products of surfaces: some new examples verifying Voisin’s conjecture." Rendiconti del Circolo Matematico di Palermo Series 2 68, no. 2 (August 16, 2018): 419–31. http://dx.doi.org/10.1007/s12215-018-0367-5.

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19

Lambert, Claude. "De la nécessité de Désordre dans la Démocratie." Acta Europeana Systemica 6 (July 12, 2020): 41–48. http://dx.doi.org/10.14428/aes.v6i1.56803.

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Dans cet article, je propose dem'appuyer sur une évaluation de dysfonctionnement démocratique partagée par de nombreux citoyens. Pour ce faire, je propose d'appliquer la conjecture de Heinz Von Foerster à la société contemporaine. Laconjecturede von Foerster décrit le rapport de causalité circulaire entre une totalité (par exemple, une collectivité humaine) et ses éléments (les individus qui la composent). Elle établit que plus les relations inter-individuelles sont "rigides"plus le comportement de la totalité apparaîtra aux élémentsindividuels qui la composentcomme doté d'une dynamique propre qui échappe à leur maîtrise alors qu'elle sera d'autant plus prédictible par un observateur extérieur.C'est ainsi que je montrerai comment les relations entre les acteurs de nos sociétés sontdominées par le paradigme de l'échange marchand basé sur la quantification, l'équivalence et la liberté des acteurs. D'un point de vue éthique, on ne peut que se féliciter de la liberté ainsi permise. Au-delà du contenu de ce type de relation, je fais l'hypothèse qu'elles ont par nature un caractère rigide et prédictible dans la forme, rencontrant ainsi les conditions d'application de la conjecture de H. von Foerster. Ceci se manifeste par un sentiment régulièrement partagé que la totalité est guidée ou manipulée par un "pouvoir obscur" ou "main invisible". A la recherche de sens, cette perception alimente et justifie les discours populistes et les théories du complot. Cependant, on ne peut se satisfaire de cette explication en invitant les acteurs de l'intérieur à adopter le point de vue de l'observateur extérieur. Les deux objectivités –de l'intérieur et de l'extérieur -se valent selon le point de vue. Il est donc utile de rester au niveau du monde-vécu et d'explorer un angle de vue alternatif. C'est ainsi que l'on peut mettre en rapport la liberté individuelle permise par le marché vis-à-vis de la liberté d'engagement dans la vie sociale. Cette dernière est rencontrée au sein d'une communauté animée par le paradigme du don-contredon au sens de Marcel Mauss. Le paradigme du don se manifeste aujourd'hui dans les formes d'associations informelles: associations de voisins, réseaux d'échanges, mouvements de militance morale... La relation qui s'établi par le don a un caractère imprédictible et incertain qui tranche avec la "simplicité" et univocité de la relation marchande. Ce caractère incertain du don-contredon introduit dans la société de l'inattendu, ce qui d'un point de vue systémique est favorable à l'innovation et l'adaptation. On ne peut alors que s'inquiéter de la tendance actuelle à la marchandisation du service public, la professionnalisation du service aux personnes, à l'application des critères du marché au fonctionnement des associations du secteur non-marchand... Cette tendance représente une réduction de la complexité sociale favorable à la rigidification des relations. Il ne s'agit pas ici de faire le procès du paradigme de l'échange marchand car il assure l'indépendance et l'accès à l'étranger au réseau d'échange. Il s'agit de suggérer une multiplicité des typologies de relations. C'est ainsi que d'un point de vue systémique, on ne peut que se féliciter d'une certaine forme de désordre social. Le régime démocratique accueille par définition conflit, débat et diversité de comportements.

20

Maulik, Davesh, and Andrei Neguţ. "LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN." Journal of the Institute of Mathematics of Jussieu, August 3, 2020, 1–39. http://dx.doi.org/10.1017/s1474748020000377.

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The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$ . We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface $S$ .

21

Li, Zhiyuan, and Ruxuan Zhang. "Beauville–Voisin Filtrations on Zero-Cycles of Moduli Space of Stable Sheaves on K3 Surfaces." International Mathematics Research Notices, June 13, 2022. http://dx.doi.org/10.1093/imrn/rnac161.

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Abstract The Beauville–Voisin conjecture predicts the existence of a filtration on a projective hyper-Kähler manifold opposite to the conjectural Bloch–Beilinson filtration, called the Beauville–Voisin filtration. In [13], Voisin has introduced a filtration on zero-cycles of an arbitrary projective hyper-Kähler manifold. On the moduli space of stable objects on a projective K3 surface, there are other candidates constructed by Shen–Yin–Zhao and Barros–Flapan–Marian–Silversmith in [1, 10] and more recently by Vial in [11] from a different point of view. According to the work in [11], all of them are proved to be equivalent except Voisin’s filtration. In this paper, we show that Voisin’s filtration is the same as the other filtrations. As an application, we prove a conjecture in [1].

22

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.

23

Totaro, Burt. "THE INTEGRAL HODGE CONJECTURE FOR 3-FOLDS OF KODAIRA DIMENSION ZERO." Journal of the Institute of Mathematics of Jussieu, February 18, 2020, 1–21. http://dx.doi.org/10.1017/s1474748019000665.

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We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.

24

Fu, L. "Beauville-Voisin Conjecture for Generalized Kummer Varieties." International Mathematics Research Notices, April 7, 2014. http://dx.doi.org/10.1093/imrn/rnu053.

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25

Dan, Ananyo. "On a conjecture of Harris." Communications in Contemporary Mathematics, June 15, 2020, 2050028. http://dx.doi.org/10.1142/s0219199720500285.

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For [Formula: see text], the Noether–Lefschetz locus [Formula: see text] parametrizes smooth, degree [Formula: see text] surfaces in [Formula: see text] with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the Noether–Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large [Formula: see text], but is true for [Formula: see text]. She also showed that for [Formula: see text], there are finitely many reduced, irreducible components of [Formula: see text] of non-maximal codimension. In this paper, we prove that for any [Formula: see text], there are infinitely many non-reduced irreducible components of [Formula: see text] of non-maximal codimension.

26

Li, Zhiyuan, and Xun Zhang. "Deligne-Beilinson cohomology of the universal K3 surface." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.60.

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Abstract O’Grady’s generalised Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarised K3 surfaces. In [4], this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4th DB-cohomology group of universal oriented polarised K3 surfaces with at worst an $A_1$ -singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O’Grady’s original conjecture in DB cohomology.

27

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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Ottem, John Christian, and Fumiaki Suzuki. "An $${\mathcal {O}}$$-acyclic variety of even index." Mathematische Annalen, March 10, 2023. http://dx.doi.org/10.1007/s00208-023-02581-2.

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AbstractWe give the first examples of $${\mathcal {O}}$$ O -acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over $${\mathbb {P}}^{1}$$ P 1 such that any multi-section has even degree over the base $${\mathbb {P}}^{1}$$ P 1 and show moreover that we can find such a family defined over $${\mathbb {Q}}$$ Q . This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel–Jacobi maps.

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Laterveer, Robert, and Charles Vial. "Zero-cycles on double EPW sextics." Communications in Contemporary Mathematics, July 27, 2020, 2050040. http://dx.doi.org/10.1142/s0219199720500406.

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The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

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Villaflor Loyola, R. "Small codimension components of the Hodge locus containing the Fermat variety." Communications in Contemporary Mathematics, May 17, 2021, 2150053. http://dx.doi.org/10.1142/s021919972150053x.

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We characterize the smallest codimension components of the Hodge locus of smooth degree [Formula: see text] hypersurfaces of the projective space [Formula: see text] of even dimension [Formula: see text], passing through the Fermat variety (with [Formula: see text]). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension [Formula: see text]. Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing through Fermat, the ones associated to a complete intersection cycle of type [Formula: see text] attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether–Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.

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Claesson, Anders, and Svante Linusson. "$n!$ matchings, $n!$ posets (extended abstract)." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2817.

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International audience We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labelled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabelled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Mélou et al. and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53]. Nous montrons qu'il y a $n!$ couplages sur $2n$ points sans emboîtement (de voisins) à gauche. Nous définissons aussi un ensemble d'EPO (ensembles partiellement ordonnés) sans motif $(2+2)$ naturellement étiquetés, et montrons qu'il y a $n!$ tels EPO sur $n$ éléments. Notre travail a été inspiré par Bousquet-Mélou, Claesson, Dukes et Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. Ces auteurs donnent des bijections entre quatre classes d'objets combinatoires: couplages sans emboîtement de voisins (dû à Stoimenow), EPO sans motif $(2+2)$ non étiquetés, permutations évitant un certain motif, et des objets appelés suites à montées. Nous pensons que certaines statistiques sur nos couplages et nos EPO pourraient généraliser le travail de Bousquet-Mélou et al. et nous proposons une conjecture à ce sujet. Nous identifions aussi des sous-ensembles naturels de couplages et d'EPO qui sont énumérés par la même séquence que la classe des EPO sans motif $(2+2)$ non étiquetés. Nous donnons des bijections qui démontrent l'équivalence entre les restrictions sur les emboîtements (d'arcs voisins) et les restrictions sur les croisements (d'arcs voisins). Nous pensons que ces bijections présentent un intérêt propre. L'une de ces bijections passe par certaines matrices triangulaires supérieures à coefficients entiers qui ont été récemment étudiées par Dukes et Parviainen [Electron. J. Combin. 17 (2010) #R53].

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Konvalinka, Matjaž, and Igor Pak. "Cayley and Tutte polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AR,..., Proceedings (January 1, 2012). http://dx.doi.org/10.46298/dmtcs.3055.

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International audience Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm. Les polytopes de Cayley ont été définis récemment comme des ensembles convexes de compositions de Cayley introduits par Cayley en 1857. Dans ce papier, nous résolvons la conjecture de Braun. Cette dernière exprime le volume du polytopes de Cayley en termes du nombre de graphes connexes. Nous étendons ce résultat à des déformations de polytopes de Cayley à deux variables, à savoir les polytopes de Tutte. Le volume de ces derniers est donnè par une évaluation du polynôme de Tutte du graphe complet. Notre approche est basée sur une triangulation explicite des polytopes de Cayley et Tutte. Nous démontrons que les simplexes de ces triangulations correspondent à des arbres marqués. La pierre angulaire de notre démonstration est une bijection directe basées sur l'algorithme de la recherche du premier voisin sur le graphe.

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Laterveer, Robert. "Zero-cycles on self-products of varieties: some elementary examples verifying Voisin’s conjecture." Bollettino dell'Unione Matematica Italiana, September 17, 2020. http://dx.doi.org/10.1007/s40574-020-00259-0.

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Fink, Simon, Eva Ruffing, Tobias Burst, and Sara Katharina Chinnow. "Emotional citizens, detached interest groups? The use of emotional language in public policy consultations." Policy Sciences, May 14, 2023. http://dx.doi.org/10.1007/s11077-023-09508-3.

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AbstractIn public consultations, policymakers give stakeholders access to the policymaking process in exchange for technical or political information. Our article proposes to analyze not only the policy positions, but the emotional content of consultation contributions. In our descriptive study, we explore two conjectures: First, citizens contributions to public consultations display more emotions than contributions by corporate actors, and second, contributions mentioning concrete policies display more emotions than contributions referring to the abstract policy framework. We use dictionary-based sentiment coding to analyze ~ 7300 contributions to the consultation of German electricity grid construction planning. Our analysis shows that citizens’ contributions contain more emotional terms, especially voicing fear. Moreover, if contributions refer to a specific power line, they contain less joy, but more fear and sadness. Thus, we show a way to conceptualize and measure the link between public policies and the emotions they trigger.