Non-closed Fields Research Articles

Two Cycle Class Maps on Torsion Cycles

Abstract We compare two cycle class maps on torsion cycles and show that they agree up to a minus sign. The first one goes back to Bloch [6], with recent generalizations to non-closed fields. The second is the étale motivic cycle class map $\alpha ^{i}_{X}\colon \operatorname{CH}^{i}(X)_{{\mathbb{Z}}_{\ell }}\to H^{2i}_{L}(X,{\mathbb{Z}}_{\ell }(i))$ restricted to torsion cycles.

Derived Equivalence for Elliptic K3 Surfaces and Jacobians

Abstract We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners, which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

Rationality of Fano threefolds over non-closed fields

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank~1 over an arbitrary field of zero characteristic.

Compactifications of the affine plane over nonclosed fields

In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an application, we have a criterion for del Pezzo fibrations with canonical singularities whose generic fibers are not smooth to contain vertical $\mathbb{A}^2$-cylinders.

RATIONALITY OVER NONCLOSED FIELDS OF FANO THREEFOLDS WITH HIGHER GEOMETRIC PICARD RANK

AbstractWe prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$ . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

Invariant Brauer group of an abelian variety

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.

Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505).We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2.We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.

Rationality of even-dimensional intersections of two real quadrics

We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.

On birational automorphisms of Severi-Brauer surfaces

The generators of the group of birational automorphisms of any Severi-Brauer surface non-isomorphic over an algebraically non-closed field to the projective plane are explicitly described.

Rationality of complete intersections of two quadrics over nonclosed fields

We study rationality problems for smooth complete intersections of two quadrics.We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a nonclosed field.

Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over$\mathbb {Q}$and$\mathbb {F}_q(t)$, and conclude with a pair of hyperkähler 4-folds over$\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

The Clemens–Griffiths method over non-closed fields

We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $\bar{k}$-rational but not $k$-rational. When $k=\mathbb{R}$, we can moreover ensure that their real locus is diffeomorphic to the real locus of a smooth projective $\mathbb{R}$-rational variety and that all their unramified cohomology groups are trivial.

Signature morphisms from the Cremona group over a non-closed field

We prove that the plane Cremona group over a perfect field with at least one Galois extension of degree 8 is a non-trivial amalgam, and that it admits a surjective morphism to a free product of groups of order 2.

O‐Plane Backreaction and Scale Separation in Type IIA Flux Vacua

AbstractWe construct AdS4 flux vacua of type IIA string theory in the supergravity (large volume, small ) regime, including the backreaction of O6‐planes. Our solutions are the localized versions of the smeared solutions on Calabi‐Yau orientifolds studied by DeWolfe, Giryavets, Kachru and Taylor and in other works. We find that the O‐plane backreaction in these solutions generates warping, a varying dilaton, non‐closed RR field strengths and internal curvature. Just like their smeared counterparts, the localized solutions admit stabilized moduli, a parametric control over string corrections and a parametric separation between the AdS and KK scales. Our explicit expressions furthermore make precise the common lore that smeared solutions should approximate the exact ones in the large‐volume, small‐ limit. Finally, our solutions appear to violate a recent swampland conjecture about an absence of scale separation in supersymmetric AdS vacua. We make a simple observation explaining why this happens in these solutions in contrast to most other AdS solutions in string theory.

On the Action of the Restricted Weyl Group on the Set of Orbits a Minimal Parabolic Subgroup

We construct the action of the restricted Weyl group on the set of principal families of orbits of a minimal parabolic subgroup over an algebraically nonclosed field. Additionally, we relate this action to the action on a polarized cotangent bundle. These results generalize the corresponding results of Knop on the action of the Weyl group on the families of Borel orbits of maximal complexity and rank.

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

On complexity of reductive group actions over algebraically non-closed field and strong stability of the actions on flag varieties

We annonce the results generalizing the Vinberg's Complexity Theorem for the action of reductive group on an algebraic variety over algebraically non-closed field. Also we give new results on the strong k-stability for the actions on flag varieties.

On the Complexity of Reductive Group Actions over Algebraically Nonclosed Field and Strong Stability of the Actions on Flag Varieties

We annonce the results generalizing the Vinberg's Complexity Theorem for the action of reductive group on an algebraic variety over algebraically non-closed field. Also we give new results on the strong k -stability for the actions on flag varieties.

Cox rings over nonclosed fields

We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.

Auroras in the cusp and its poleward vicinity: a case study

Summary We present a case study of the dayside aurora observed simultaneously with optical instruments from the ground and with auroral particle spectrometers aboard the DMSP F16 and F17 satellites. Optical observations were carried out with an all-sky camera at the Polar Geophysical Institute (PGI) observatory Barentsburg on Svalbard. The aurora as a whole moved equatorward in response to negative turning of the IMF Bz component and then the distinct faint rayed arc intensified, moved to the north and faded. Satellite DMSP F17 crossed the cusp twenty minutes after Bz turned southward. Joint analysis of optical and satellite data showed that faint auroral structures are embedded into the cusp precipitations and correspond to the bursts of electron precipitations with energy below 100 eV. The next satellite crossed the camera field-of-view ten minutes later and the data showed that the source of the faded poleward moving rayed arc was located, most probably, on the non-closed magnetic field lines. This finding and the presence of ion-energy dispersion in the DMSP data allows us to make the conclusion that the dayside reconnection may be considered as the reason for this kind of aurora activity. In this study we also estimated the altitude and horizontal scale of auroral rays in the cusp.