Extremal rational elliptic threefolds

Weak Fano threefolds with del Pezzo fibration

This article treats smooth weak Fano 3-folds having an extremal ray of type D. Smooth weak Fano 3-folds with an extremal ray of type D except of degree 6 are classified into 47 deformation types.

On stable rationality of Fano threefolds and del Pezzo fibrations

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Rationality of rationally connected threefolds admitting non-isomorphic endomorphisms

We prove a structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds, or more generally for threefolds with big anti-canonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be rational. As a consequence, we show (without using the classification) that every smooth Fano threefold having a non-isomorphic surjective endomorphism is rational.

Stable rationality in smooth families of threefolds

We exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.

Rationality of $\mathbb{Q}$-Fano threefolds of large Fano index

We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.

实法诺三维簇的一个合理性准则

A rationality criterion for real Fano threefolds

该研究探讨光滑几何有理法诺三维簇实轨迹的连通性，提出并证明了一个实合理性的充分准则，为相关几何对象的合理性分析提供了理论依据。

On Examples of Supersingular and Rationally Chain Connected Threefolds

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.

Non-rationality of the symmetric sextic Fano threefold

We give a simple proof of the non-rationality of the Fano threefold defined by the equations Σx_i = Σx_i^2 = Σx_i^3 = 0 in P^6 .

Rationality of Fano threefolds with terminal Gorenstein singularities, I

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

Rationality of Fano threefolds over non-closed fields

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.

Finiteness of rational curves of degree 12 on a general quintic threefold

We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.

On the rational Enriques-Fano threefolds discovered by Fano

It was Fano who first classified Enriques-Fano threefolds. However his arguments appear to contain several gaps. In this paper, we will verify some of his assertions through the use of modern techniques.

Rationality in families of threefolds

We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is the union of at most countably many closed subfamilies.

The spaces of rational curves on del Pezzo threefolds of degree one

We prove the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank $1$ and degree $1$. As corollaries, we confirm Geometric Manin's conjecture and enumerativity of certain Gromov-Witten invariants for these threefolds.

On relative rational chain connectedness of threefolds with anti-big canonical divisors in positive characteristics

In this paper we prove two results about the rational chain connectedness for klt threefolds with anti-big canonical divisors in the relative setting.

具有Gorenstein终端奇点的Fano三维簇上的有理曲线

Rational curves on Fano threefolds with Gorenstein terminal singularities

本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。

具有一维纤维的三重极值收缩的一般大象：例外情形

General elephants for threefold extremal contractions with one-dimensional fibers: exceptional case

本文研究具有一维纤维的三重极值收缩，在每分支至多含一个指数大于2点的条件下，证明一般成员是具有 Du Val 奇点的正规曲面。

Rational Curves on Calabi-Yau Threefolds: Verifying Mirror Symmetry Predictions

In this paper, the numbers of rational curves on general complete intersection Calabi-Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry.

On Extremal Contractions from Threefolds to Surfaces: the Case of One non-Gorenstein Point

Let $f\colon X\to S$ be an extremal contraction from a threefold with only terminal singularities to a surface. We study local analytic structure such contractions near degenerate fiber $C$ in the case when $C$ is irreducible and $X$ has on $C$ only one non-Gorenstein point.

On Rational Connectedness of Globally F-Regular Threefolds

In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.

Rationality of Fano threefolds with terminal Gorenstein singularities, II

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

Codimension one regular foliations on rationally connected threefolds

In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.

The cone conjecture for some rational elliptic threefolds

We verify the Morrison--Kawamata conjecture for a certain class of rational threefolds, namely blowups of P^3 in the base locus of a net of quadrics with no reducible members. This seems to be the first verified case of the conjecture for klt Calabi--Yau pairs in dimension 3 with non-zero boundary divisor.

The cone of moving curves of a smooth Fano-threefold

In this short note we show that the closed cone of moving curves of a smooth Fano-threefold is polyhedral. The proof relies on a famous result of Bucksom, Demailly, Paun and Peternell which says that the strongly movable cone is dual to the cone of pseudoeffective divisors. Finally, we relate the extremal rays of the cone of moving curves on a threefold with extremal rays in the cone of strongly movable curves on a birational model obtained by a Mori contraction.

具有 Artin-Mumford 有理性障碍的光滑二次三重覆盖的双重覆盖

Double covers of smooth quadric threefolds with Artin-Mumford obstructions to rationality

本文分类了作为光滑二次三重流形双重覆盖且具有 Artin-Mumford 有理性障碍的节点 Fano 三重流形，证明其属于三个明确描述的族，其中两个族为首次发现。

Two rational nodal quartic threefolds

We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.

关于某些 Fano 三维簇的有理性

On the rationality of certain Fano threefolds

本文研究具典范奇点的 Gorenstein Fano 三维簇有理性，证明三角、含平面、高亏格及含普通直线等四类情形在相应阈值下均为有理簇。

Rational curves on prime Fano threefolds of index 1

We study the moduli spaces of rational curves on prime Fano threefolds of index 1. For general threefolds of most genera we compute the dimension and the number of irreducible components of these moduli spaces. Our results confirm Geometric Manin's Conjecture in these examples and show the enumerativity of certain Gromov-Witten invariants.

Derived-equivalent rational threefolds

We describe an infinite set of smooth projective threefolds that have equivalent derived categories but are not isomorphic, contrary to a conjecture of Kawamata. These arise as blow-ups of $\mathbb P^3$ at various configurations of 8 points, which are related by Cremona transformations.

Stable rationality of orbifold Fano threefold hypersurfaces

We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.

Reducible Calabi-Yau threefolds with countably many rational curves

We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.

Towards the Classification of Weak Fano Threefolds with ρ= 2

In this paper, we provide examples of Sarkisov links of type II between complex projective Fano threefolds $X$ with $ρ(X) = 1$. To show examples of these links, we study smooth weak Fano threefolds with extremal rays of type $E$. We further assume that the pluri-anticanonical morphism contracts only a finite number of curves. We numerically classify smooth weak Fano threefolds with $ρ(X) = 2$ having an extremal ray of type $E$ both before and after a flop as well as prove the existence of some of these numerical cases.

Extremal Bundles on Calabi-Yau Threefolds

We study constructions of stable holomorphic vector bundles on Calabi-Yau threefolds, especially those with exact anomaly cancellation which we call extremal. By going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds. We then introduce a generalized version of Hartshorne-Serre construction and use it to yield extremal bundles of general ranks and study their geometry. In light of this probing the geometry of the space of stable vector bundles, we revisit the DRY conjecture on stable r…

Maximum number of $\mathbb{F}_q$-rational points on nonsingular threefolds in $\mathbb{P}^4$

We determine the maximum number of $\mathbb{F}_q$-rational points that a nonsingular threefold of degree $d$ in a projective space of dimension $4$ defined over $\mathbb{F}_q$ may contain. This settles a conjecture by Homma and Kim concerning the maximum number of points on a hypersurface in a projective space of even dimension in this particular case.

On smooth rational threefolds of P^5 with rational non-special hyperplane section

It is known that the smooth rational threefolds of P^5 having a rational non-special surface of P^4 as general hyperplane section have degree d=3,... ,7. We study such threefolds X from the point of view of linear systems of surfaces in P^3, looking in each case fosr an explicit description of a birational map from P^3 to X. For d=3,..., 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing-down o…

Linear systems representing threefolds which are scrolls on a rational surface

Let X be a scroll over a rational surface. We construct a linear system of surfaces in P^3 yielding a birational map from P^3 to X. We apply this construction to the scrolls of Bordiga and Palatini.

Rational curves of degree 10 on a general quintic threefold

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle O(-1)^2. Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components in F.

Rational Curves in Calabi-Yau Threefolds

We study the set of rational curves of a certain topological type in general members of certain families of Calabi-Yau threefolds. For some families we investigate to what extent it is possible to conclude that this set is finite. For other families we investigate whether this set contains at least one point representing an isolated rational curve.

Moduli spaces of rational curves on Fano threefolds

We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases. The key to our approach is Geometric Manin's Conjecture which predicts the number of components parameterizing free curves.

叶状三维流形的均匀有理多胞体与全局ACC

Uniform rational polytopes of foliated threefolds and the global ACC

本文证明了维数≤3且具函数边界的叶状结构存在均匀有理lc多胞体，并以此证明了具任意DCC系数的叶状三维流形的全局ACC，同时给出了lc阈值累积点的应用。

On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.

有理拓扑可缩仿射三维空间上的向量丛

Vector Bundles on Rational Topologically Contractible Affine Threefolds

研究有理连通且维数为3的拓扑可缩光滑仿射复簇上代数向量丛是否平凡的问题，通过证明给出肯定回答，如证明科拉斯 - 拉塞尔三维空间上代数向量丛是平凡的。

关于度数为$22$的素Fano三维流形中有理四次曲线的一个注记

A remark on rational quartic curves in prime Fano threefolds of degree $22$

利用度22的素Fano三维流形$V_{22}$与五次del Pezzo三维流形$V_5$之间的Sarkisov联系，证明$V_{22}$中有理四次曲线的Hilbert方案到射影空间$\mathbb{P}^4$存在一个一般2对1的有理映射。

理性Fano锥面三维流形的自同构群与线性化

Automorphism groups and linearizability of rational Fano conic bundle threefolds

本文研究了Fano锥面三维流形的自同构群与线性化问题，证明了No. 2.18类Fano三维流形对于其自同构群的线性化性质。

某些实三维叠的有理性研究

On the rationality of some real threefolds

本文研究某些实三维二次曲面包络在实数域及更一般的实闭域上的有理性，通过未分枝上同调和birational刚性技术，得出正负结果及具体构造。

$(\mathbb{Z}/4)^4$对有理连通三维流形的作用

Actions of $(\mathbb{Z}/4)^4$ on rationally connected threefolds

研究$(\mathbb{Z}/4)^4$对有理连通三维流形的作用，证明相关双有理结论，得出该群不嵌入$\operatorname{Cr}_3(\mathbb{C})$并结合早期结果，实现对$(\mathbb{Z}/m)^r$嵌入情况的完整分类。

Extremal metrics on del Pezzo threefolds

We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variety is equal to 1/2.

Rational curves on generic quintic threefolds

Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over complex numbers and $C_0$ be an irreducible rational curve on $X_0$. Let $$c_0: \mathbf P^1\to C_0\subset X_0$$ be its normalization. In this paper, we show (1) $c_0$ must be an immersion, i.e. the differential $(c_0)_\ast: T_{t}\mathbf P^1 \to T_{c_0(t)} X_0$ is injective at each $t\in \mathbf P^1$, (2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0/X_0})=0.$$

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r general points of a smooth elliptic quartic curve B of P^3 and let L be any line bundle of X. The aim of this paper is to provide an explicit algorithm for determining the dimension of H^0(X,L).

On extremal contractions from threefolds to surfaces: the case of one non-Gorenstein point and non-singular base surface

In this paper we study a local structure of extrmal contractions $f\colon X\to S$ from threffolds $X$ with only terminal singularities onto a surface $S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein point on a fiber we prove that either the linear system $|-K_X|$, $|-2K_X|$ or $|-3K_X|$ contains a "good" divisor.

度$22$素Fano三维簇上低次有理曲线的Hilbert概型

Hilbert schemes of low-degree rational curves on a prime Fano threefold of degree $22$

本文完整描述了度$22$素Fano三维簇上次数不超过$6$的有理曲线的Hilbert概型，利用二割线圆锥曲线几何，并导出DT型不变量。

Rational curves and strictly nef divisors on Calabi--Yau threefolds

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ν(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.

Non-rational nodal quartic threefolds

The $\mathbb{Q}$-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is $\mathbb{Q}$-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not $\mathbb{Q}$-factorial if and only if it contains a plane. However, there are non-rational non-$\mathbb{Q}$-factorial nodal quartic 3-folds in $\mathbb{P}^4$. In particular, we prove the non-rationality of a general non-$\mathbb{Q}$-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.

有理三维簇与Kummer曲面的伪自同构

Pseudo-automorphisms of rational threefolds and Kummer surfaces

本文将45个经典Kummer曲面自同构提升为$\mathbb{P}^3$沿6点15线爆破所得三维簇的伪自同构，并描述了由这些已知伪自同构生成的群的基本域。