Related papers: On minimal 4-folds of general type with $p_g \geq …

For smooth projective 3-folds of general type, we prove that the relative canonical stability $\mu_s(3)\leq 8$. This is induced from our improved result of Koll\'ar: the m-canonical map of a smooth projective 3-fold of general type is…

We establish the Noether inequality \[\textrm{Vol}(X)\geq \frac{4}{3}p_g(X)-\frac{10}{3}\] for all projective $3$-folds $X$ of general type with geometric genus $5\leq p_g(X)\leq 10$ where $\textrm{Vol}(X)$ is the canonical volume. This…

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of…

This note describes minimal surfaces $S$ of general type satisfying $p_g\geq 5$ and $K^2=2p_g$. For $p_g\geq 8$ the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of…

In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.

We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with…

Let $S$ be a minimal surface of general type with $p_g=0$ and $K^2=6$, such that its bicanonical map $\fie\colon S\to\pp^6$ is not birational. The map $\fie$ is a morphism of degree $\le 4$ onto a surface. The case of $\deg\fie=4$ is…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

In this note we provide a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known bound of 72.

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. Furthermore, the ample model $V$ achieving the minimal volume is determined uniquely up to isomorphism.…

In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion…

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $\chi (\omega_X)$ and $p_g(X)$: 1) Assume that the canonical map $\Phi_{|K_X|}$ is of fiber type. Let $F$ be a smooth model…

By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical…

In this note it is shown that, given a smooth minimal complex surface of general type S with p_g(S)=0, K^2_S=3, for which the bicanonical map is a morphism, then the degree of the bicanonical map of S is not equal to 3. This completes our…

Answering a question posed by Enriques, we construct a minimal smooth algebraic surface $S$ of general type over the complex numbers with $K^2 = 45$ and $p_g = 4$, and with birational canonical map. Our surface is a regular (q=0) ball…

We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we…

In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.

The "canonical dimension" of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical…

In this paper we classify completely all regular minimal surfaces with K^2=8, p_g=4 whose canonical map is composed with an involution. We obtain six unirational families of respective dimensions 28,28,32,33,38,34. The last two are…

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of…