For rational points on algebraic varieties defined over a number field K, we study the behavior of the property of weak approximation with Brauer–Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension $\mathit{L}|\mathit{K}$ such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals $\mathbb{Q}$ or certain quadratic number fields. We give an explicit example when $\mathit{K}=\mathbb{Q}$.

We generalize Bestvina’s notion of a $\mathcal{Z}$-boundary for a group to that of a “coarse $\mathcal{Z}$-boundary”. We show that established theorems about $\mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $\mathcal{Z}$-boundaries become theorems when applied to coarse $\mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $\mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a “model $\mathcal{Z}$-geometry”. In accordance with the existing theory, we also develop an equivariant version of the above—that of a “coarse $\mathit{E}\mathcal{Z}$-boundary”.

In this paper, we collect various structural results to determine when an integral homology 3-sphere bounds an acyclic smooth 4-manifold, and when this can be upgraded to a Stein manifold. In a different direction, we study whether a smooth embedding of connected sums of lens spaces in ${\mathbb{C}}^2$ can be upgraded to a Stein embedding, and we have determined that this never happens.

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian.

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs, we consider general types of multivariate Bernstein–Sato ideals associated to arbitrary factorizations of our germ. We show that these ideals are principal, and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we obtain nice estimates for the zero locus of its Bernstein–Sato ideal for arbitrary factorizations and show that the Bernstein–Sato ideal attached to a factorization into linear forms is reduced. As an application, we independently verify and improve upon an estimate of Maisonobe regarding standard Bernstein–Sato ideals for reduced, generic arrangements: we compute the Bernstein–Sato ideal for a factorization into linear forms, and we compute its zero locus for other factorizations.

Generalizing the involution length of the complex hyperbolic plane, we obtain that the α-length of $\mathrm{PU}(2,1)$ is 4, that is, every element of $\mathrm{PU}(2,1)$ can be decomposed as the product of at most four special elliptic isometries with parameter α. We also describe the isometries that can be written as the product of two or three such special elliptic isometries.

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in ${\mathbb{R}}^{\mathit{d}}$. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

Let $\mathcal{A}\subset \mathcal{B}$ be integral domains. Suppose $\mathcal{A}$ is Noetherian and $\mathcal{B}$ is a finitely generated $\mathcal{A}$-algebra. Denote by $\bar{\mathcal{A}}$ the integral closure of $\mathcal{A}$ in $\mathcal{B}$. We show that $\bar{\mathcal{A}}$ is determined by finitely many unique discrete valuation rings. Our result generalizes Rees’ classical valuation theorem for ideals. We also obtain a variant of Zariski’s main theorem.

An Artin algebra Λ is said to be of finite Cohen–Macaulay type if, up to isomorphism, there are only finitely many indecomposable modules in $\mathcal{G}(\mathrm{\Lambda })$, the full subcategory of $\mathrm{mod}\mathrm{\Lambda }$ consisting of all Gorenstein projective (right) Λ-modules. In this paper, we study 1-Gorenstein algebras of finite Cohen–Macaulay type through $\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$, the category of finitely presented $\mathcal{G}(\mathrm{\Lambda })$-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for ${\mathit{T}}_3(\mathrm{\Lambda })$, the 3 by 3 lower triangular matrices over Λ, to be of finite Cohen–Macaulay type. Finally, the structure of almost split sequences will be described explicitly in a special subcategory of $\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$, denoted by ${\mathit{\vartheta }}^{-1}(\mathcal{G}(\mathrm{\Lambda }))$. If Λ is self-injective, ${\mathit{\vartheta }}^{-1}(\mathcal{G}(\mathrm{\Lambda }))=\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$.

We prove that some Gromov–Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use the Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in ${\mathbb{P}}^3$ and ${\mathbb{P}}^5$ are computed. The results are consistent with existing results.