Conic configurations via dual of quartic curves

Explicit equations of 800 conics on a Barth-Bauer quartic

We present equations of $800$ conics on a Mukai's smooth quartic which is an explicit model of a K3 surface recently considered by Alex Degtyarev. The number $800$ is a current record for the number of irreducible conics on a smooth quartic in ordinary projective space.

圆锥线性系与平面四次曲线束

Conic linear series and pencils of plane quartics

研究光滑复曲线\(C\subset\mathbb{P}^{3}\)上由固定次数锥切出的线性系，通过系统研究其族的相关性质，应用该研究证明了存在特定的非等平凡四次曲线束。

Rationality of real conic bundles with quartic discriminant curve

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with ir…

具有锥纤维化的四次del Pezzo曲面的Manin猜想

Manin's conjecture for quartic del Pezzo surfaces with a conic fibration

建立了具有锥丛结构的非奇异四次del Pezzo曲面上有界高的有理点个数的渐近公式。

有理二次曲线的四次del Pezzo曲面上有理点的计数

Counting rational points on quartic del Pezzo surfaces with a rational conic

上，下界，预期的数量级，得到的数量有理点的有界高度上的任何四次del Pezzo表面上的$\mathbb{Q}$，其中包含一个圆锥曲线定义在$\mathbb{Q}$。

与13个结点的四次曲面接触的二次曲线

Conics Touching a Quartic Surface with 13 Nodes

受扭量空间构造问题的启发，研究了与四次曲线相切的二次曲线的参数空间。对于复$\PP ^3$中的一个给定的实四次曲面，研究了这个参数空间，该曲面恰好有13个普通结点作为奇点。特别是，它的不可约成分进行了说明。

弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何学 via 理性椭圆面和扎里斯基对

Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs

研究弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何结构，通过理性椭圆面和扎里斯基对构造度数为7和8的扎里斯基对。

射影三维空间中Heisenberg不变四次曲面上的曲线

Curves on Heisenberg invariant quartic surfaces in projective 3-space

本文研究了在Heisenberg群$H_{2,2}$下不变的光滑四次曲面族$X \subset \mathbb{P}^3$.对于一个非常一般的这样的曲面$X$，我们证明了$X$的Picard数为16，并确定了它的Picard群.原来，一般海森堡不变四次曲线包含320个光滑二次曲线，在非常一般的情况下，这些二次曲线的集合生成皮卡德群。

关于具有最大数量最大切线的四次曲线

On quartics with the maximal number of the maximal tangency lines

本文研究了Fermat和Komiya-Kuribayashi四次曲线上最大切线的直线排列与点构型，并进一步分析了其六切点及相关圆锥曲线构型。

四次 del Pezzo 曲面上的圆锥曲线 10-网

On the 10-web by conics on the quartic del Pezzo surface

本文研究四次 del Pezzo 曲面上的圆锥曲线 10-网，证明 Bol 网及其阿贝尔恒等式的显著性质可推广至该网及其超对数关系。

具有普通奇点的四次二重体

Quartic double solids with ordinary singularities

本文研究了具有双二次曲线的四次曲面上的射影三维空间的双覆盖的三重分支的第三个同调群上的混合Hodge结构。我们处理的Torelli问题，这样的三倍。

Zariski multiples associated with quartic curves

We investigate Zariski multiples of plane curves $Z_1, \dots, Z_N$ such that each $Z_i$ is a union of a smooth quartic curve, some of its bitangents, and some of its 4-tangent conics. We show that, for plane curves of this type, the deformation types are equal to the homeomorphism types, and that the number of deformation types grows as $O(d^{62})$ when the degree $d$ of the plane curves tends to infinity.

关于某些特殊平面四次曲线的Waring分解的注记

A remark on Waring decompositions of some special plane quartics

本文研究一类高阶平面四次线的Waring分解。主要结果如下。设x，l_1，.，L_7是三维向量空间上线性型，Q是二次型。如果x^2q=l_1^4+.+ l_7^4和线l_1=0，...，P^2中的l_7=0与x=0相交于七个不同的点，则直线x=0（可能不正确地）与二次曲线q=0相切。

Coble四次曲线、$S_6$不变四次三次曲线和Wiman-Edge六次曲线

Coble fourfold, $S_6$-invariant quartic threefolds, and Wiman-Edge sextics

我们构造了Coble四重（四维射影空间在Igusa四次线上分支的双重覆盖）奇点的两个小分解。我们用它们证明了所有的$S_6$不变的三维四次曲线都对五次del Pezzo曲面上的圆锥丛是双有理的，并给出了Wiman-Edge束的判别曲线。作为应用，我们检验了$S_6$ -不变的三维四元数是单有理的，得到了其中四种特殊四元数的有理性和其它四元数的非有理性的新证明，并将它们的Weil除子类群描述为$S_6$ -表示.

网的故事：del Pezzo四次曲面上的二次曲线网与旋量十重态的Gelfand-MacPherson网

A story of webs: the webs by conics on del Pezzo quartic surfaces and Gelfand-MacPherson's web of the spinor tenfold

本文通过显式计算证明，del Pezzo四次曲面上的二次曲线网与Bol网的相似性源于后者与旋量十重态的Gelfand-MacPherson网的相似性，后者是Bol网的秩5推广，含可推导四次del Pezzo曲面权3超对数恒等式的2-阿贝尔关系。

四次曲线及其镜像对偶的运动Zeta函数

Motivic Zeta Functions of the Quartic and its Mirror Dual

我们使用Bultot公式计算四次曲线K3的复曲面退化及其Gross-Siebert镜像对偶退化的动机zeta函数。我们检查这个明确的例子，单值和镜像对偶莱夫谢茨算子的对数的识别工作在积分水平。

$(\mathbb{Z}/2\mathbb{Z})^{4}$不变四次K3曲面族的Picard群

The Picard Group of Various Families of $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant Quartic K3 Surfaces

本文的主题是研究在一定的$(\mathbb{Z}/2\mathbb{Z})^{4}$作用下保持不变的各种四次K3曲面族。特别是，我们描述的家庭，其一般成员包含$8,16,24$或$32$线，以及$320$二次曲线发现的埃克隆德（其中一些退化成上述线）。本文的后半部分致力于寻找这些家族的一般成员的皮卡德群，并将其描述为一个格。事实证明，对于每一个家庭的皮卡德群的一个非常一般的表面是产生的线和二次曲线躺在说表面。

平面四次线：二重线矩阵

Plane quartics: the matrix of bitangents

Aronhold的经典结果指出，一个平面四次线可以恢复的配置任何Aronhold系统的双切线，即特殊的7元组的双切线，使六个点，其中任何subtriple的双切线接触四次线不在于同一个圆锥曲线在射影平面。在2005年，Lehavi证明了一个光滑的平面四次曲线可以从它的28个二重切线中显式地重建出来，这一结果改进了Aronhold的恢复曲线的方法。在2011年的一篇论文中，Plaumann，Sturmfels和Vinzant引入了一个8 × 8的对称矩阵，参数化了一个非奇异平面四次曲线的双切线。他们的建设的出发点是黑塞的结果，其中每一个光滑的四次曲线正好有36个等价类的线性对称行列式表示。本文从平面四次曲线的28条双切线出发，解决了双切线矩阵的逆问题。

Coble四次超曲面的自对偶性及其应用

Self-duality of Coble's quartic hypersurface and applications

亏格为3的非超椭圆曲线C上具有固定平凡行列式的半稳定秩为2的向量丛的模空间M_0同构于P^7中的一个四次超曲面（Coble的四次曲面）。我们证明了M_0是自对偶的，并且它的极映射将稳定丛E \in M_0与丛F联系起来，丛F的特征为dim H^0（C，E \otimes F）= 4.在射影空间PH ^0（C，E \otimes F）上配置一个二次曲面网E，证明了E的平面四次Hessian曲线的同构类E \in M_0所对应的映射是亏格为3的曲线的模空间的支配映射.

On the cuspidal locus in the dual varieties of Segre quartic surface

Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.

三次三重函数θ divisor的新模型

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。

关于平面实曲线与M-四次曲线的相互安排

On mutual arrangements of a plane real curve relative to an $M$-quartic with an oval-snake

本文研究了平面实曲线与M-四次曲线的相互安排，证明了在特定条件下，实曲线与四次曲线的拓扑等价性，并探讨了伪全纯曲线的类似结论。

A special configuration of $12$ conics and generalized Kummer surfaces

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $…

关于两条圆锥曲线的交点

On the Intersection of Two Conics

本文针对两条圆锥曲线求交问题，提出两种坐标变换方法：一为将一条曲线化为抛物线以解四次方程，二为利用自极三角形将问题简化为二次方程求解。

与仿射型秩2簇突变共轭的双有理映射及其几何

Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry

生成$A^{(2)}_2$型簇代数的簇变量的突变约化为由四次双有理映射给出的二维离散可积系统。该映射的不变曲线是一条奇异的四次曲线，它的奇异分解在二次曲线上导出一个离散的可积系统，该系统由共轭于$A^{(2)}_2$型簇突变的三次双有理映射控制.证明了二次曲线也是由$A^{(1)}_1$型簇突变所产生的二次双有理映射的不变曲线，并且二次曲线上的两个双有理映射是可交换的.最后，将交换双有理映射归结为$A^{(1)}_1$型离散Toda格的谱曲线所产生的椭圆曲线上点的加法的奇异极限.

关于可约平面曲线通过Galois覆盖的补的拓扑

On the topology of the complements of reducible plane curves via Galois covers

设${\mathcal {B}}$是可约的约化平面曲线。引入了一种新的观点，通过Galois覆盖和Alexander多项式来研究$(\PP^2, {\mathcal {B}})$的拓扑.我们证明了它的有效性，通过例子Zagliki $N$ -plets圆锥和圆锥四次配置。

圆锥丛三维流形上的曲线类及其在有理性中的应用

Curve classes on conic bundle threefolds and applications to rationality

研究几何有理曲面上圆锥丛三维流形的曲线类结构，构造Galois等变同态推广Beauville结果，并应用于精细中间Jacobian挠子(IJT)有理性阻碍，证明在低算术复杂度基域下IJT可刻画有理性。

由直线和圆锥连通簇覆盖

Covered by Lines and Conic Connected Varieties

研究了直线覆盖的嵌入簇的一些性质，给出了通过其两个一般点的奇异二次曲线存在的一个数值判据。我们表明，我们的标准是尖锐的。圆锥连接，覆盖线，QEL，LQEL，总理Fano，缺损和双重缺损品种密切相关。利用Ein和Zak的著名结果研究了上述几类对象之间的关系。

Segre quartic surfaces and minitwistor spaces

Segre surfaces in the title mean quartic surfaces in $\mathbb{CP}^4$ which are the images of weak del Pezzo surfaces of degree four under the anti-canonical map. We first show that minimal minitwistor spaces with genus one are exactly Segre quartic surfaces. By a kind of Penrose correspondence, Zariski open subsets of the projective dual varieties of these surfaces admit Einstein-Weyl structure. We investigate structures of these dual varieties in detail. In particular, we determine the degrees of these varieties (namely the classes of the Segre surfaces), as well as structure of several compo…

支撑在四次曲线上的稳定层的模空间的Chow环

Chow ring of the moduli space of stable sheaves supported on quartic curves

受S. Katz，我们计算了Hilbert多项式$4m+1$在射影平面上的稳定层的模空间的Chow环和上同调环.作为副产品，我们得到了所有线丛的全陈类和欧拉示性数，为平面上的奇异对偶性提供了数值数据。

圆锥曲面映射与圆锥

Conics Quadrics Mapping & Cones

本文提出通过圆锥和二次曲面的射线获取五点圆锥和九点二次曲面的隐式方程的方法，并展示如何利用双四元数运动映射将共面五点置于圆锥上，同时探讨圆锥对称性对解的生成影响。

Quartic del Pezzo surfaces with a Brauer group of order 4

We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a rational point. We also answer a question of Várilly-Alvarado and Viray by showing that the Brauer groups these surfaces cannot be vertical with respect to any projection away from a plane. We conclude that the available techniques for proving existence of rational points or even Zariski density do not directly apply if there is no Brauer-Manin obstruction …

Kummer quartic surfaces, strict self-duality, and more

In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $γ$ on the minimal resolution $S$ of $X$. Then we study the corresponding Enriques surfaces $S/ γ$. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefalú quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves through an isogeny with k…

K3多面体及其四次曲面

K3 Polytopes and their Quartic Surfaces

K3多面体出现在热带四次曲面的补曲面中。它们是标准四面体第四膨胀中自反多面体的正则幺模中心三角剖分的对偶。探索这些组合的对象，我们分类K3多面体高达$30$顶点。他们的号码是$36\,297\,333$。研究了热带化为K3多胞形的四次曲面的奇异轨迹。这些曲面在几何不变理论意义下是稳定的。

On rationally integrable planar dual and projective billiards

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others. In the paper we study its following generalized dual version stated by S.Tabachnikov. Consider a closed smooth strictly convex curve $γ\subset\mathbb{RP}^2$ eq…

Quaternary quartic forms and Gorenstein rings

A quaternary quartic form, a quartic form in four variables, is the dual socle generator of an Artinian Gorenstein ring of codimension and regularity 4. We present a classification of quartic forms in terms of rank and powersum decompositions which corresponds to the classification by the Betti tables of the corresponding Artinian Gorenstein rings. This gives a stratification of the space of quaternary quartic forms which we compare with the Noether-Lefschetz stratification. We discuss various phenomena related to this stratification. We study the geometry of powersum varieties for a general f…

关于有理可积平面对偶多重台球与分段光滑射影台球

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

本文分类了分段C⁴光滑的有理0次齐次可积射影台球，并通过对偶多重台球证明其积分最小次数可为2、4或12。

同调稳定性与四次 del Pezzo 曲面上有理曲线的 Manin 猜想

Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces

本文证明了有理曲线到分裂四次 del Pezzo 曲面的 Manin 猜想和 Cohen--Jones--Segal 猜想，通过 bar 复形、维数估计和虚拟高度 zeta 函数等方法，验证了 Batyrev 等人的启发式方法。

Sextactic points on the Fermat cubic curve and arrangements of conics

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to $F$, studying higher degree curves passing through higher order division points on $F$, studying…