Related papers: Smooth 3-dimensional canonical thresholds

We show the abundance theorem for arithmetic klt threefold pairs whose closed point have residue characteristic greater than five. As a consequence, we give a sufficient condition for the asymptotic invariance of plurigenera for certain…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of…

We say that a topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open sets $\{U_n\mid n\in\omega \}$ of $X$, we can find $x_n\in U_n$ such that the sequence $(x_n)$ has no convergent subsequences.…

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an…

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain…

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not…

We show, in this first part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $16$. We produce examples with…

Let $X$ be a K3 surface, let $C$ be a smooth curve of genus $g$ on $X$, and let $A$ be a line bundle of degree $d$ on $C$. Then a line bundle $M$ on $X$ with $M\otimes\mathcal{O}_C=A$ is called a lift of $A$ . In this paper, we prove that…

A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the…

This note aims to improve known numerical bounds proved earlier by Chen \cite{PAMS} and Chen-Hacon \cite{Chen-Hacon} and to present some new examples of smooth minimal 3-folds canonically fibred by surfaces (resp. curves) of geometric genus…

A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for…

We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and…

We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the $\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang in the…

We consider Fano threefolds $X$ with canonical Gorenstein singularities. Under additional assumption that $X$ has at least one non-cDV point we prove a sharp bound of the degree: $-K_X^3\le 72$.

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the…

A singularity is said to be exceptional (in the sense of V. Shokurov), if for any log canonical boundary, there is at most one exceptional divisor of discrepancy -1. In our previous paper (math.AG/9805004) we found two examples of…

We prove that a general rational smooth Fano threefold admits a toric model. More precisely, for a general rational smooth Fano threefold $X$, we show the existence of a boundary divisor $D$ for which $(X,D)\simeq_{\rm cbir}…

We consider a family of surfaces of general type $S$ with $K_S$ ample, having $K^2_S = 24, p_g (S) = 6, q(S)=0$. We prove that for these surfaces the canonical system is base point free and yields an embedding $\Phi_1 : S \rightarrow…

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with…