相关论文: Normality of general elephants on 3-fold terminal …

Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f : (X, C) \to (Z, o)$ such that $C = f^{-1} (o)_{\mathrm{red}}$ and $-K_X$ is $f$-ample. Assume that…

We study deformations of a pair of a $\mathbb{Q}$-Fano $3$-fold $X$ with only terminal singularities and its elephant $D \in |{-}K_X|$. We prove that, if there exists $D \in |{-}K_X|$ with only isolated singularities, the pair $(X,D)$ can…

Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities…

We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.

We give a classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $X$ possess a regular involution, which acts freely on some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a morphism which…

Consider a singular curve $\Gamma$ contained in a smooth 3-fold $X$. Assuming the general elephant conjecture, the general hypersurface section $\Gamma\subset S\subset X$ is Du Val. Under that assumption, this paper describes the…

We prove that if $X---> X^+$ is a threefold terminal flip, then $c_1(X).c_2(X)\leq c_1(X^+).c_2(X^+)$ where $c_1(X)$ and $c_2(X)$ denote the Chern classes. This gives the affirmative answer to a Question by Xie \cite{Xie2}. We obtain the…

Let X be the germ of a Gorenstein 3-fold singularity and C a smooth curve through it such that the general hyperplane section S of X containing D is DuVal of type E6 or E7. In this paper we obtain criteria for the existence of a terminal…

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\to (Z,o)$ such that $C=f^{-1}(o)_{\red}$ and $-K_X$ is ample. Assume that a general member…

Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two…

We study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities…

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic…

In 2009, de Fernex and Hacon proposed a generalization of the notion of the singularities to normal varieties that are not Q-Gorenstein. Based on their work, we generalize Kleiman's transversality theorem to subvarieties with log terminal…

In order to find useful information to complete the classification of Enriques-Fano threefolds, we will computationally study the singularities of some known Enriques-Fano threefolds of genus 6, 7, 8, 9, 10, 13 and 17. We will also deduce…

This is a revised version of my paper which appeared in J. Math. Sci. Univ. Tokyo. We prove that for a flipping contraction from a Gorenstein terminal 4-fold, the pull back of a general hyperplane section of the down-stair has only…

Let X be a smooth threefold. We show that if $X_i\dashrightarrow X_{i+1}$ is a flip which appears in the $K_X$-MMP, then $c_1(X_i)^3-c_1(X_{i+1})^3$ is bounded by a constant depending only on $b_2(X)$.

We study Nash valuations on 3-fold terminal singularities, especially in type cAx/2. We find that, in type cAx/2, exceptional prime divisors computing the minimal discrepancy (which is 1/2 in this case) induce Nash valuations. We conjecture…

Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the optimal inequality: $$K_X^{3}\geq \frac{4}{3}\chi(\omega_X)-2,$$ where $\chi(\omega_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the dualizing sheaf…

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following…