We study the variation of linear sections of hypersurfaces in . We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree hypersurface in varies maximally for . In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to -planes in projective space and to free rational curves on arbitrary varieties.
We describe separating -invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for -invariants of several copies of the algebra of octonions in case of a field of odd characteristic.
We study a family of exponential sums that arises in the study of expanding horospheres on . We prove an explicit version of general purity and find optimal bounds for these sums.
We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.
We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple -values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that -conjugate rooted tree maps are their antipodes.
We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants . In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to , and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to modulo . Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.
The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least when the degree is a prime . This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular.
We prove that the Galois equidistribution of torsion points of the algebraic torus extends to the singular test functions of the form , where is a Laurent polynomial having algebraic coefficients that vanishes on the unit real -torus in a set whose Zariski closure in has codimension at least . Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of .
Given a smooth projective toric variety of Picard rank 2, we resolve the diagonal sheaf on by a linear complex of length consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.
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