The goal of this note is to study a conjectural picture on lower bounds of Seshadri constants of indecomposable polarized abelian varieties. This is inspired by some ideas of Debarre on the subject and the author's previous work on syzygies of abelian threefolds using the convex geometry of Newton–Okounkov bodies.

By previous work of Cedó, Smoktunowicz, and Vendramin, one already knows that the analog of Grün's lemma fails to hold for perfect skew braces when the socle is used as an analog of the center of a group. In this paper, we use the annihilator instead of the socle. We shall show that the analog of Grün's lemma holds for perfect two-sided skew braces but not in general.

A prequantization bundle is a circle bundle over a symplectic surface with negative Euler class. A connection 1-form induces a natural contact form on such a bundle. The purpose of this paper is to study symplectic embeddings of balls into the associated disk bundles. To this end, we compute the ECH cobordism maps induced by these disk bundles. In addition, we compute the ECH spectrum of the prequantization bundles over the sphere and the torus.

Let 𝑝 be a prime number. By a result of Ozaki, the capitulations of ideals in ℤ_𝑝-extensions and the finite submodules of Iwasawa modules are closely related. In this article, we discuss this relationship in ℤ_𝑝^𝑑-extensions.

We study the embedded topology of certain conic-line arrangements of degree 7. Two new examples of Zariski pairs are given. Furthermore, we determine the number of connected components of the realization spaces of the conic-line arrangements with prescribed combinatorics. We also calculate the fundamental groups using SageMath and the package Sirocco in the appendix.

Let 𝑘₀ be an algebraic number field of finite degree, 𝑆₀ be a finite set of primes and 𝐿_{𝑆₀} be the field obtained by adjoining to 𝑘₀ all primitive 𝑞-th roots of unity, where 𝑞 runs over all primes not belonging to 𝑆₀. We shall consider, for an odd prime 𝑙, the maximal unramified pro-𝑙 abelian extension of 𝐿_{𝑆₀} and investigate the structure of this Galois group with certain cyclotomic action.

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.

For a resolution space (\tilde{𝑋}, 𝐸) of a normal complex surface singularity (𝑋, 𝑜), the fundamental cycle 𝑍_𝐸 and maximal ideal cycle 𝑀_𝐸 are important geometric objects associated to (𝑋, 𝑜), which satisfy 𝑀_𝐸 ≧ 𝑍_𝐸. In 1966, M. Artin proved that 𝑀_𝐸 = 𝑍_𝐸 for all resolutions of all rational singularities. However, for non-rational singularities, it is a delicate problem whether 𝑀_𝐸 = 𝑍_𝐸 or not. Any normal surface double point (i.e., multiplicity two) is a hypersurface singularity defined by 𝑧² = 𝑓(𝑥, 𝑦). For such singularities, we prove that 𝑀_𝐸 > 𝑍_𝐸 holds on the minimal resolution if and only if 𝑓 has a canonical decomposition 𝑓 = 𝑓_[𝐿] 𝑓_[𝑐] 𝑓_[𝑜] in ℂ{𝑥, 𝑦} called “Laufer decomposition”. Moreover, we give a numerical procedure to determine whether 𝑀_𝐸 = 𝑍_𝐸 or not on the minimal resolution from the embedded topology of the branch curve singularity ({𝑓 = 0}, 𝑜).

Given a complex variety 𝑋, suppose there is a degeneration 𝔛 of it to a union of toric varieties glued along toric divisors. Then, one can use tropical curves to study holomorphic curves on 𝑋 through degenerate curves on the central fiber 𝑋₀ of 𝔛. In such studies, the case corresponding to so-called non-superabundant tropical curves is relatively well understood. This is the case where the curve satisfies a version of transversality. On the other hand, in the case corresponding to superabundant tropical curves, where transversality fails, not much is known. The first important step for such a study is describing the obstruction to deforming degenerate curves on 𝑋₀ to generic fibers of 𝔛. In this paper, we present a general formula describing such an obstruction. In the study of superabundant tropical curves, allowing higher-valent vertices (i.e., those with valency greater than three) is essential, although there have been few studies of them in this context. Our formula covers such cases.

In this paper, we consider the linearized compressible Navier–Stokes equations with non-slip boundary conditions in the half space ℝ^𝑁₊. We prove the generation of a continuous analytic semigroup associated with this compressible Stokes system with non-slip boundary conditions in the half space ℝ^𝑁₊ and its 𝐿₁ in time maximal regularity. We choose the Besov space ℋ^𝑠_𝑞,𝑟 = 𝐵^𝑠+1_𝑞,𝑟 (ℝ^𝑁₊) × 𝐵^𝑠_𝑞,𝑟(ℝ^𝑁₊)^𝑁 as an underlying space, where 1 < 𝑞 < ∞, 1 ≤ 𝑟 < ∞, and −1 + 1/𝑞 < 𝑠 < 1/𝑞. We prove the generation of a continuous analytic semigroup {𝑇(𝑡)}_{𝑡 ≥ 0} on ℋ^𝑠_𝑞,𝑟, and show that its generator admits maximal 𝐿₁ regularity. Our approach is to prove the existence of the resolvent in ℋ^𝑠_𝑞,1 and some new estimates for the resolvent by using 𝐵^𝑠+1_𝑞,1(ℝ^𝑁₊) × 𝐵^{𝑠 ± 𝜎}_𝑞,1(ℝ^𝑁₊) norms for some small 𝜎 > 0 satisfying the condition −1 + 1/𝑞 < 𝑠 − 𝜎 < 𝑠 < 𝑠 + 𝜎 < 1/𝑞.

Let 𝑀 be a surface with a Riemannian metric and 𝑈𝑀 the unit tangent bundle over 𝑀 with the canonical contact sub-Riemannian structure 𝐷 ⊂ 𝑇(𝑈𝑀). In this paper, the complete local classification of singularities, under the Legendre projection 𝑈𝑀 → 𝑀, is given for sub-Riemannian geodesics of (𝑈𝑀, 𝐷). Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre projection from 𝑈𝑀 to the space of Riemannian geodesics on 𝑀. The duality on Legendre singularities is observed related to the pendulum motion.

Consider a 𝑑-dimensional branching Brownian motion starting with a single particle at the origin and let 𝑛_𝑡 be the number of particles at time 𝑡 whose ancestral lines have remained up to 𝑡 within a ball of radius 𝑟(𝑡) centered at the origin, where 𝑟(𝑡) increases sublinearly with 𝑡. We obtain a full limit large-deviation result as time tends to infinity on the probability that 𝑛_𝑡 is atypically small. A phase transition is identified, at which the nature of the optimal strategy to realize the aforementioned large-deviation event changes, and the Lyapunov exponent giving the decay rate of the associated large-deviation probability is continuous. As a corollary, we also obtain a kind of law of large numbers for 𝑛_𝑡 under the stronger assumption that 𝑟(𝑡) increases subdiffusively with 𝑡.