Gloire Grace Bockondas, Samuel Boissiere
The present paper deals with lines contained in a smooth complex cubic threefold. It is well-known that the set of lines of the second type on a cubic threefold is a curve on its Fano surface. Here we give a description of the singularities of this curve.
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Eckardt points on a cubic threefold
本文综述光滑复三次三重形上的Eckardt点，提出计算全部Eckardt点的方法，并构造了不含Eckardt点但含三重线的三次三重形。
Rationality of singular cubic threefolds over $\mathbb R$
本文研究实数域上奇异三次三倍体的有理性性质，为相关代数几何问题提供新的理解。
In this paper, we classify groups which faithfully act on smooth cubic threefolds. It turns out that there are exactly $6$ maximal ones and we describe them with explicit examples of target cubic threefolds.
Morphisms from Quintic Threefolds to Cubic Threefolds are Constant
证明了从四维射影空间中的5次超曲面到四维射影空间中的非奇异3次超曲面的每一个态射必然是常数。在这个过程中，我们还分类的态射从射影平面的非奇异三次三倍的3次多项式。
Equivariant geometry of singular cubic threefolds, II
本文研究具有非结点孤立奇点的三次三维簇上有限群作用的线性化问题，属于奇异三次三维簇等变几何研究的延续部分。
A new proof of the nonrationality of cubic threefolds
本文给出了一般三次三重曲线的非有理性的一个新证明：如果一般三次三重曲线是有理的，则相应的中间雅可比矩阵是曲线雅可比矩阵的乘积。我们退化一个通用的三次三倍到塞格雷三次三倍，所以有一个退化的中间雅可比以及。与退化的Prym家族相关联的是幺模向量系统。一般立方三重的合理性意味着幺模系统将是余图切割。然而，我们证明，得到的幺模系统是一个众所周知的对称非余图切割称为E_5。
Equivariant geometry of cubic threefolds with non-isolated singularities
本文研究有限群在具有非孤立奇点的三次三重流形上作用的可线性化性问题，属于等变几何范畴。
A categorical invariant for cubic threefolds
我们证明了三次三倍的Torelli定理的一个分类版本。更精确地说，我们表明，非平凡的一个半正交分解的导出范畴的三次三重特征的同构类。
Abel-Jacobi maps associated to smooth cubic threefolds
提出了有理连通簇上的空间参数化有理曲线的一些问题。我们给出了三次三重数的部分解答。我们的许多结果以前证明了Iliev，Markushevich和Tikhimirov通过分析三次三重上的向量丛。我们的证明使用剩余而不是向量丛。
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
Stable equivariant birationalities of cubic and degree 14 Fano threefolds
本研究提出Pfaffian-Grassmannian对应的等变版本，以此构造出三次三维簇与14次Fano三维簇间非平凡的扭稳定等变双有理等价实例，为等变双有理几何提供新案例。
The Hasse principle for 9-nodal cubic threefolds
本文证明了在数域上具有9个奇点的三次三流形满足哈塞原理，解决了该类代数簇的局部-整体原则问题。
Equivariant geometry of singular cubic threefolds
本文借助上同调工具、中间雅可比簇、Burnside不变量与等变极小模型纲领，研究有限群在奇异三次三重流形上作用的可线性化问题。
Lines on cubic threefolds and fourfolds containing a plane
该研究刻画了特征非2域上含平面的一般三次三维簇的直线Fano概形，以此刻画非闭域上这类三次三维簇的有理性，并显式构造含平面三次四维簇直线Fano簇的拉格朗日纤维化。
On Fourier-Mukai type autoequivalences of Kuznetsov components of cubic threefolds
本文确定了光滑复三次超曲面的Kuznetsov分量的所有Fourier-Mukai型自等价群，并为该范畴的Torelli定理的Fourier-Mukai版本提供了新证明。
Curves of small degree on cubic threefolds
我们描述了Hilbert方案参数化曲线的三次三倍的程度最多为5。在即将发表的论文中，我们使用这种描述给出了一个新的证明和推广定理的伊利耶夫，Markushevich和Tikhimirov。
We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular, we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some of the current research in algebraic geometry related to Segre's work.
It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.
The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant
我们证明了三重Koras-Russell三次仿射上圆柱的Makar-Limanov不变量是平凡的。这意味着在加性群对这个簇的所有代数作用下不变的正则函数是常数。
Hyperplane sections of cubic threefolds
本文证明光滑三次超曲面的超平面截面可涵盖一般三次曲面，建立了三次曲面与三次超曲面之间的结构联系。
We give an algorithm to compute the zeta function of the Fano surface of lines of a smooth cubic threefold $F$ into $\mathbb{P}^4$ defined over a finite field. We obtain some examples of Fano surfaces with supersingular reduction.
A remark on the Abel-Jacobi morphism for the cubic threefold
设$X$是一个光滑三次三重集，$J(X)$是它的中间雅可比矩阵.证明了存在一个余维2圈$Z$，且$Z_{t}$对每个$t\in J(X)$都是同调平凡的，使得Abel-Jacobi映射诱导的态射$ϕ_{Z}: J(X)\rightarrow J(X)$是恒等式.这肯定地回答了一个问题Voisin的情况下，立方三倍。
We study the geometry of quintic threefolds $X\subset \mathbb{P}^4$ with only ordinary triple points as singularities. In particular, we show that if a quintic threefold $X$ has a reducible hyperplane section then $X$ has at most $10$ ordinary triple points, and that this bound is sharp. We construct various examples of quintic threefolds with triple points and discuss their defect.
A note on Fano surfaces of nodal cubic threefolds
我们研究的Picard品种的Fano表面的节点和轻度尖点三次在任意特征的因子Fano表面上的因子对称产品的曲线亏格4。
$E_6$-local systems from cubic threefolds
研究通过构造立方三项式三种fold模空间上的无穷多个局部系统，其代数单个群为例外群$E_6$，并基于阿贝尔 étale 覆盖的芬诺方案中位上同调得出主要贡献。
Given a cubic hypersurface $X\subset \mathbb{P}^4$, we study the existence of Pfaffian representations of $X$, namely of $6\times 6$ skew-symmetric matrices of linear forms $M$ such that $X$ is defined by the equation $Pf(M)=0$. It was known that such a matrix always exists whenever $X$ is smooth. Here we prove that the same holds whenever $X$ is singular, hence that every cubic threefold is Pfaffian.
Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.
On the Tate conjecture for the Fano surfaces of cubic threefolds
在P^4中光滑三次三重X的Fano曲面将X上的线参数化。本文证明了特征不为2的素域上的有限型域上的Fano曲面满足Tate猜想。
The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14
研究了光滑三次三重曲面上的椭圆五次曲面族和有理四次曲面族的Abel-Jacobi映射。证明了它们的类纤维对于五次函数是5维射影空间，对于四次函数是关于三次函数自身的光滑的3维双有理簇。本文是Markushevich-Tikhomirov最近工作的继续，他证明了第一个Abel-Jacobi映射通过三次三重空间上稳定秩为2的向量丛的模分量进行因子分解，其Chern数为$c_1=0, c_2=2$，由Serre从椭圆五次空间构造而得，并且从模空间到中间Jacobian的因子分解映射是不规则的.上面的结果意味着étale映射的度是1，因此向量丛的模分量对于中间Jacobian是双有理的。作为应用，证明了14次Fano簇的周期映射的类纤维对于相应的三次三重Fano簇的中间Jacobian是双有理的.
On the classification of singular cubic threefolds
本文分类复三次三流形上孤立奇点的组合，推广三次曲面的经典结果，并给出简单奇点构型与特定图子图对应的组合刻画。
Arithmetic Torelli maps for cubic surfaces and threefolds
人们早就知道，对于一个复杂的三次曲面或三次曲面，人们可以正则地把一个主要极化的阿贝尔簇联系起来。我们给出了一个适用于算术基上的三次函数的构造。这回答了，远离素数2，德利涅的一个老问题和库德拉和拉波波特最近的一个问题。我们进一步分类的Mumford-Tate群的阿贝尔簇出现，并给出额外的算术应用。
We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category which does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.
On some Moduli spaces of stable vector bundles on cubic and quartic threefolds
本文研究了三次和四次三重上秩为二的稳定向量丛的模空间。在许多情况下，我们证明了模空间是完备的和不可约的，并且一般的模空间有消失的中间上同调。在一种情况下，除了模空间的一个分量之外，所有的分量都有这样的向量丛。
The moduli space of cubic threefolds via degenerations of the intermediate Jacobian
Clemens和Griffiths的一个著名的结果是，光滑三次三重可以从其中间雅可比矩阵恢复。本文讨论了这类阿贝尔簇的可能退化，从而给出了用这种方法得到的三次三重模空间的紧化的描述。这种紧化与Allcock-Carlson-Toledo和Looijenga-Swierstra的工作中构造的紧化之间的关系也被考虑，并且在精神上类似于低亏格曲线的模空间的各种紧化之间的关系。
The Fano surface of the Klein cubic threefold
证明了Klein三次三重曲面$F$是唯一一个自同构为11阶的光滑三次三重曲面。计算了$F$的中间雅可比矩阵的周期格，并研究了它的Fano曲面$S$ .我们还计算了一组纤维化的$S$到曲线的正属和这些纤维化的纤维之间的交叉。这些纤维生成Néron-Severi群的一个指数为2的子群，我们得到了这个群的一组生成元。$S$的Néron-Severi群有秩$25=h^{1,1}$和判别式$11^{10}$。
Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrising lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper we show that if X is generic, the Fano surface is the only surface of minimal class in J.
Special subvarieties in the locus of intermediate Jacobians of cubic threefolds
本文研究五次主极化阿贝尔簇模空间中与三次三维簇中间雅可比相关的特殊子簇，给出充分条件并发现新的正维特殊子簇例子。
The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to…
We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two were previously left as open numerical possibilities, namely $(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold wi…
G-birationally rigid cubic threefolds
本文研究了三次三次曲面与有限子群的组合，探讨了在G作用下三次三次曲面的理性刚性性质及主要贡献。
A new model for the theta divisor of the cubic threefold
本文给出了一般三次三重$X$的中间雅可比矩阵的θ因子的双有理模型。我们使用的标准实现的$X$作为一个圆锥束和平面四次曲线的$4-$维家庭是完全相切的判别五次曲线这样的圆锥束结构。该系列曲线上的偶数θ特性的附加数据为我们提供了θ因子的模型。
Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold
本文证明了Edoukou关于非退化Hermitian三重超曲面与三次三重超曲面公共点最大数的猜想在d=3且q≥7时成立。
The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree 5 and a ball quotient
本文研究了费马三次曲面的Fano曲面S。证明了S是5次del Pezzo曲面的81次阿贝尔覆盖，S上12条不相交的椭圆曲线之并的补是球商.这个球商的格与Deligne-Mostow格数1有关。
The period map for cubic threefolds
Allcock-Carlson-Toledo定义了一个三次三折的周期图，它取10维球商的值。Voisin的一个定理意味着这是一个开嵌入。我们确定它的图像，并表明，在代数水平上，这相当于一个识别的代数SL（5，C）-不变多项式的第三对称功率的对偶重言式表示SL（5，C）与明确描述的代数的亚纯自守形式的复杂的10-球。
Vector bundles on the cubic threefold
设X是光滑三次三重空间，M是X上稳定秩为2的向量丛的模空间，其行列式为平凡行列式，c_2=2（该空间非空的最小值）.最近Druel，Iliev，Markushevich和Tikhomirov的结果导致了M及其Maruyama紧化M '的精确描述：后者同构于X沿Fano曲面爆破的中间Jacobian。在这篇综述论文中，我们解释了这些结果，并讨论了一些应用。
Number of triple points on complete intersection Calabi-Yau threefolds
本文研究射影及加权射影空间中 Calabi–Yau 三维簇的普通三重点数量，证明二次–四次完全交至多有10个，并给出六次超曲面的精确界与实例。
Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds
本文证明光滑三次三维流形上由一对不相交直线决定的希尔伯特概型不可约分支光滑且同构于对角线上直线法诺簇对称积的爆破，并研究其与超平面截面几何及布里奇兰德模空间的关系。
We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.
Irrationality of the first stabilization of a complex smooth cubic threefold
本文证明所有光滑复三次三维簇与射影直线乘积的非理性，并构造一个稳定有理但非理性的 Fano 簇，其高次幂与对称平方均为有理。
The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold
研究了一般三重三次曲面上椭圆五次曲面族的Abel-Jacobi映射。证明了它通过三次三次空间上稳定秩为2的向量丛的模分支因子分解，其中Chern数c_1=0，c_2=2，其通点表示由椭圆五次曲线通过Serre构造得到的一个向量丛.映射到模空间一点的椭圆五次线在希尔伯特曲线格式内的5维射影空间中变化，并且从模空间到中间雅可比矩阵的映射是不确定的。作为辅助结果，证明了椭圆正规五次函数族和有理正规四次函数族在一般三次三重函数上的不可约性。这意味着所考虑的模量分量的唯一性。利用Clemens-Griffiths和Welters的技巧计算无穷小Abel-Jacobi映射。
It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6 x 6 skew-symmetric matrix of linear forms in P^4. To compactify the space of such Pfaffian representations of a given cubic and to study the construction in families as well as for singular or reducible cubics, it is thus natural to consider the incidence correspondence of Pfaffian representations inside the product of the space of semistable skew-symmetric 6 x 6 matrices of linear forms in P^4 and the space of cubics. Here we describe concretely the irreducible component of this incidence correspondenc…
The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.
We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6, both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkaehler fourfolds of K3^{[2]}-type…
The defect of a cubic threefold and applications to intermediate Jacobian fibrations
本文研究具有孤立奇点的三次三流形的缺陷不变量，利用过奇点的直线曲线拓扑数据计算缺陷，刻画混合霍奇结构，并证明非Q-阶乘性等价于包含平面或三次直纹面，进而联系到三次四流形的中间雅可比纤维化。
The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$
研究 $\mathcal A_5$ 中 Siegel 度量限制在三次三维流形的中间雅可比轨迹上的第二基本形式，证明其像包含于某个乘法映射的核中。
Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds
P^（d-1）中具有两个平面三次曲线纤维化的（1，d）-极化阿贝尔曲面位于两个椭圆P^2-涡卷中。这些涡卷的并集形成一个可约的卡-丘三重。在本文中，我们表明，这发生在d<10，并分析家庭这样的表面和3倍详细时，d=6。特别是，可约的卡-丘3-折叠变形，在这种情况下，不可约的非正常奇异性。
Cubic threefolds and abelian varieties of dimension five
本文证明了Mumford定理的逆命题：设$A$是一个5维主极化阿贝尔簇，其θ因子有唯一的奇点，并假定该奇点的重数为3.然后$A$是同构的主要极化阿贝尔簇的中间雅可比光滑三重。证明方法是分析$A$的θ因子的可能奇异性，并最终证明$A$是可能奇异的平面五次曲线的Prym簇.
We study the moduli space of pairs $(X,H)$ consisting of a cubic threefold $X$ and a hyperplane $H$ in $\mathbb P^4$. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of $K3$ type and, on the other hand, the study of the singularity $O_{16}$ (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating $(X,H)$ to certain "lattice polarized" cubic fourfolds $Y$. A period map for the pairs $(X,H)$ is then defined using the periods o…
For $X$ a smooth cubic threefold we study the Plücker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a unique family of curves. The analysis also shows that there is a unique pentagonal connecting family of curves, which leads to the fact that the connecting gonality of $S$ is five whereas the degree of irrationality, i.e.\ the minimal degree of a rational map from $S$ to $\mathbb{P}^2$, is six.
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