Freeness of adjoint linear systems on threefolds with non Gorenstein Q-factorial terminal singularities or some quotient singularities

非Gorenstein的极小Fano三维簇的反canonical次数

Degrees of non-Gorenstein canonical Fano threefolds with Picard number one

研究了Picard数为一的非Gorenstein Q-极小canonical Fano三维簇的反canonical次数上界，得出最佳上界为200/3。

A Minimal Model Program for $\mathbb{Q}$-Gorenstein varieties

The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal mode…

On classification of Q-Fano 3-folds with Gorenstein index 2 and Fano index 1/2

We obtain a classification of a Q-factorial Q-Fano 3-fold $X$ with the following properties: the Picard number of $X$ is 1; the Gorenstein index of $X$ is 2; the Fano index of $X$ is 1/2; $h^0 (-K_X) \geq 4$; there exists an index 2 point $P$ such that $(X,P)\simeq (\{xy +f(z^2,u)=0\} / \Bbb Z_2 (1,1,1,0), o)$ with $ord f(Z,U)=1$.

Picard 数为一且具有 $\mathbb{K}^*$ 作用、轨道商为 $\mathbb{P}_2$ 的 Gorenstein Fano 三维簇

Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$

本文分类了一类具有 $\mathbb{K}^*$ 作用和最大轨道商 $\mathbb{P}_2$ 的非环、$\mathbb{Q}$-因子、Gorenstein、Picard 数为一的 Fano 三维簇。

A classification of terminal quartic 3-folds and applications to rationality questions

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces wi…

On the canonical degrees of Gorenstein threefolds of general type

Let $X$ be a Gorenstein minimal projective $3$-fold with at worst locally factorial terminal singularities. Suppose that the canonical map is generically finite onto its image. C. Hacon showed that the canonical degree is universally bounded by $576$. We improved Hacon's universal bound to $360$. Moreover, we gave all the possible canonical degrees of $X$ if $X$ is an abelian cover over $\mathbb{P}^3$ and constructed all the examples with these canonical degrees.

On the Nash problem for terminal threefolds of type $cA/r$

We study Nash valuations and essential valuations of terminal threefolds of type $cA/r$. If $r=1$ or the given threefold is $\mathbb Q$-factorial, then all the Nash valuations and essential valuations can be completely described. We construct non-Gorenstein or non-$\mathbb Q$-factorial counter examples for the Nash problem.

The topology of terminal quartic 3-folds

In this thesis, I determine a bound on the defect of terminal Gorenstein quartic 3-folds. More generally, I study the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 and genus at least 3. I state a geometric "motivation" of non Q-factoriality in the case of quartics.

Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…

非拟F-分裂的正则仿射四维折面在每种特征中都存在

Non-quasi-$F$-split canonical affine fourfolds exist in every characteristic

该研究在每种特征中构造了非拟F-分裂的正则Q-因子化格罗恩斯坦仿射四维折面。

终端和典范 Gorenstein Fano 四折叠的反典范体积

Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds

本文证明终端和典范 Gorenstein Fano 四折叠的反典范体积上界，并给出尖锐界及特殊情形下的改进。

Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X) and its idempotent completion, we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to …

代数可积伴随叶状结构的极小模型纲领

Minimal model program for algebraically integrable adjoint foliated structures

本文针对代数可积伴随叶状结构证明了锥定理、收缩定理与flip存在性，推导出极小模型纲领，并证明了基点自由定理及准dlt修改的存在性。

The number of vertices of a Fano polytope

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

On the Picard number of singular Fano varieties

Let X be a Q-factorial Gorenstein Fano variety. Suppose that the singularities of X are canonical and that the locus where they are non-terminal has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D < 9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there exists a finite morphism from X to S x Y, where S is a surface with rho_S at most 9. As an application we prove that, if X has dimension 3, then rho_X is at most 10.

Boundedness of Fano threefolds with log-terminal singularities of given index

We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The new ideas of the paper include the following. 1. Using Alexeev Minimal Model program with suitable boundary to find horizontal extremal contractions. 2. Using Kollár's effective Base Point Freeness theorem. 3. Using Kawamata's result on the length of extremal curves with suitable boundary to avoid gluing curves in some cases.

Effective non-vanishing for Fano weighted complete intersections

We show that Ambro-Kawamata's non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify Ambro-Kawamata's conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface. For the proofs, we introduce the arithmetic notion of regular pairs and enlighten some interesting connection with the Frobenius coin probl…

Classification and Properties of Hyperconifold Singularities and Transitions

This paper is a detailed study of a class of isolated Gorenstein threefold singularities, called hyperconifolds, that are finite quotients of the conifold. First, it is shown that hyperconifold singularities arise naturally in limits of smooth, compact Calabi--Yau threefolds (in particular), when the group action on the covering space develops a fixed point. The Z_n-hyperconifolds---those for which the quotient group is cyclic---are classified, demonstrating a one-to-one correspondence between these singularities and three-dimensional lens spaces L(n,k), which occur as the vanishing cycles. Th…

由 Freudenthal 三元系定义的素 $\mathbb{Q}$-Fano 三维流的关键簇

Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Freudenthal triple systems

本文研究反典范余维数为4的复素 $\mathbb{Q}$-Fano 三维流的分类，提出利用 Freudenthal 三元系（FTS）中的严格正则元素条件来描述并构造其关键簇，并成功构建了具有特定奇点的仿射簇及 No.20544 素 $\mathbb{Q}$-Fano 三维流实例。