Related papers: Weak approximation for Fano complete intersections…

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such…

Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture is compatible for products of rationally…

In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also…

Let $X, Y \subset \mathbb{C}^{2n-1}$ be $n$-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point $x \in X$ to a point $y \in Y$. This set is defined as…

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let…

Let $d\geq 3$ be a constant and let $F$ be a $d$-regular graph on $[n]$ with not too many symmetries. By the union bound, the probability threshold for the existence of a spanning subgraph in $G(n,p)$ isomorphic to $F$ is at least…

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case…

Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this…

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we…

We give an approximate algorithm of computing holonomic systems of linear differential equations for definite integrals with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping…

First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian…

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has…

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a…

We show that under rather general circumstances, the almost everywhere pointwise inequality $|f|(x) \le Mf (x)$ is equivalent to a weak form of the Lebesgue density theorem, for totally bounded closed sets. We derive both positive and…

In this paper we prove that for $n$-dimensional smooth $l$-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for $l$ is equal to $\lceil \log_2(n+2) \rceil-1 .$ We also…

Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$ for every curve $C \subset X$, then $\rho_X=1$.

We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prior…

Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes…