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Algebra & Number Theory
Algebra & Number Theory
Nguyen, Van C.; Witherspoon, Sarah
For the class of examples in Section 5 of the article in question, the proof of finite generation of cohomology is incomplete. We give here a proof of existence of a polynomial subalgebra needed there. The rest of the proof of finite generation given by the authors then applies.
Schreieder, Stefan
We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve, with the exception of two cases, the stable rationality problem for any very general complex projective quadric surface bundle over [math] , given by a symmetric matrix of homogeneous polynomials. Both exceptions degenerate over a plane sextic curve, and the corresponding double cover is a K3 surface.
Orr, Martin
Let [math] be a reductive group defined over [math] and let [math] be a Siegel set in [math] . The Siegel property tells us that there are only finitely many [math] of bounded determinant and denominator for which the translate [math] intersects [math] . We prove a bound for the height of these [math] which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of [math] , and has applications to the Zilber–Pink conjecture on unlikely intersections in Shimura varieties.
¶ In addition we prove that if [math]...
Izquierdo, Diego
In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields in terms of projective hypersurfaces of small degree and Milnor [math] -theory. We establish these conjectures for finite extensions of [math] and [math] , and we prove new local-global principles over number fields and global fields of positive characteristic in the context of Kato and Kuzumaki’s conjectures.
Scarponi, Danny
By means of the theory of strongly semistable sheaves and the theory of the Greenberg transform, we generalize to higher dimensions a result on the sparsity of [math] -divisible unramified liftings which played a crucial role in Raynaud’s proof of the Manin–Mumford conjecture for curves. We also give a bound for the number of irreducible components of the first critical scheme of subvarieties of an abelian variety which are complete intersections.
Goodearl, Kenneth R.; Huisgen-Zimmermann, Birge
For any truncated path algebra [math] of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties [math] of the [math] -modules with fixed dimension vector [math] . In this situation, the components of [math] are always among the closures [math] , where [math] traces the semisimple sequences with dimension vector [math] , and hence the key to the classification problem lies in a characterization of these closures.
¶ Our first result concerning closures actually addresses arbitrary basic finite-dimensional algebras over an algebraically closed field. In the general case, it corners the closures [math] by...
Bell, Jason; León Sánchez, Omar; Moosa, Rahim
A differential-algebraic geometric analogue of the Dixmier–Moeglin equivalence is articulated, and proven to hold for [math] -groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if [math] is a commutative affine Hopf algebra over a field of characteristic zero, and [math] is an Ore extension to which the Hopf algebra structure extends, then [math] satisfies the classical Dixmier–Moeglin equivalence. Along the way it is shown that all such [math] are Hopf Ore extensions in the...
Meng, Xianchang
For any [math] , we study the distribution of the difference between the number of integers [math] with [math] or [math] in two different arithmetic progressions, where [math] is the number of distinct prime factors of [math] and [math] is the number of prime factors of [math] counted with multiplicity. Under some reasonable assumptions, we show that, if [math] is odd, the integers with [math] have preference for quadratic nonresidue classes; and if [math] is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with [math] always have preference...
Ford, Nicolas; Levinson, Jake; Sam, Steven V
We characterize the cone of [math] -equivariant Betti tables of Cohen–Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for “Boij–Söderberg theory for Grassmannians,” with the goal of characterizing the cones of [math] -equivariant Betti tables of modules over the coordinate ring of [math] matrices, and, dually, cohomology tables of vector bundles on the Grassmannian [math] . The proof uses Hall’s theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman’s geometric...
Lee, Dong Uk
For a Shimura variety of Hodge type with hyperspecial level at a prime [math] , the Newton stratification on its special fiber at [math] is a stratification defined in terms of the isomorphism class of the rational Dieudonné module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline tensors (“ [math] -isocrystal”). There has been a conjectural group-theoretic description of the [math] -isocrystals that are expected to show up in the special fiber. We confirm this conjecture. More precisely, for any [math] -isocrystal that is expected to appear (in a precise sense), we construct a special...
Kaliman, Shulim
We prove that the actions mentioned in the title are translations. We show also that for certain [math] -actions on affine fourfolds the categorical quotient of the action is automatically an affine algebraic variety and describe the geometric structure of such quotients.
Woo, Alexander; Wyser, Benjamin J.; Yong, Alexander
We develop interval pattern avoidance and Mars–Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup [math] acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.
Ara, Pere; Hazrat, Roozbeh; Li, Huanhuan; Sims, Aidan
We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the Cohen–Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal...
Dolgachev, Igor; Duncan, Alexander
We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective spaces over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an application, we show that every quartic del Pezzo surface over a perfect field of characteristic 2 has a canonical rational point and, thus, is unirational.
Polstra, Thomas; Tucker, Kevin
We present a unified approach to the study of [math] -signature, Hilbert–Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the [math] -signature to be viewed as the infimum of relative differences in the Hilbert–Kunz multiplicities of the cofinite ideals in a local ring.
Balkanova, Olga; Frolenkov, Dmitry
We study the first moment of symmetric-square [math] -functions at the critical point in the weight aspect. Asymptotics with the best known error term [math] were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size [math] in the asymptotic formula and show that the remainder term decays exponentially in [math] . The twisted first moment was evaluated asymptotically by Ng with the error bounded by [math] . We improve the error bound to [math] unconditionally and to [math] under the Lindelöf hypothesis for quadratic Dirichlet [math] -functions.
Küronya, Alex; Lozovanu, Victor
We study asymptotic invariants of linear series on surfaces with the help of Newton–Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton–Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton–Okounkov polygons. As an illustration of our ideas we reprove results of Ein–Lazarsfeld on Seshadri constants on surfaces.
Li, Chao; Zhu, Yihang
W. Zhang’s arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport–Zink space. In the minuscule case, Rapoport, Terstiege and Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.
Patrikis, Stefan
In previous work, we described conditions under which a single geometric representation [math] of the Galois group of a number field [math] lifts through a central torus quotient [math] to a geometric representation. In this paper, we prove a much sharper result for systems of [math] -adic representations, such as the [math] -adic realizations of a motive over [math] , having common “good reduction” properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over number fields)...
Pizzato, Marco; Sano, Taro; Tasin, Luca
We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI [math] which is Fano or Calabi–Yau, i.e., we prove that, if [math] is an ample Cartier divisor on [math] , then [math] is not empty. If [math] is smooth, we further show that the general element of [math] is smooth. We then verify the Ambro–Kawamata conjecture for any quasismooth weighted hypersurface. We also verify Fujita’s freeness conjecture for a Gorenstein quasismooth weighted hypersurface. ¶ For the proofs, we introduce the arithmetic notion of regular pairs and highlight some interesting connections with the Frobenius coin problem.