Chilean configuration of conics, lines and points

Barth–Bauer 八次曲面上的圆锥曲线

Conics on Barth--Bauer octics

本研究分析特殊Kummer八次曲面上的圆锥曲线与直线构型，给出圆锥曲线数的176条上界并找到达到该上界的唯一Mukai群作用曲面，还发现含8910条光滑圆锥曲线的二重平面。

若干直线与圆锥曲线构型及其Waldschmidt常数

Some line and conic arrangements and their Waldschmidt constants

本研究聚焦射影平面中若干直线与圆锥曲线构型的Waldschmidt常数，推导特定共线点集的常数值，结合希尔伯特多项式完成几何刻画，还描述了满足上界条件的共圆锥曲线点构型并研究了特定多点构型。

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。

Hirzebruch-type inequalities and plane curve configurations

In this paper we come back to a problem proposed by F. Hirzebruch in the 1980's, namely whether there exists a configuration of smooth conics in the complex projective plane such that the associated desingularization of the Kummer extension is a ball quotient. We extend our considerations to the so-called $d$-configurations of curves on the projective plane and we show that in most cases for a given configuration the associated desingularization of the Kummer extension is not a ball quotient. Moreover, we provide improved versions of Hirzebruch-type inequality for $d$-configurations. Finally, …

围绕卡诺定理

Around the Carnot theorem

本文研究卡诺定理及与之有关的点、线的构形，证明了构形中某些重要点位于同一条直线和同一条二次曲线上。给出了Bradley公式化的一个等价命题的证明。利用Carnot和Menelaus定理证明了Bradley提出的一个公开猜想。

On Seshadri constants and point-curve configurations

In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated…

二次曲线网中的纠缠四边形

Quadrangles entangled in conic nets

本文在特征非2的域上研究二次曲线网随四边形构型变化而保持不变的纠缠现象，证明纠缠等价于中点平分条件及平分线场相等，并给出中心轨迹决定网的结论。

什么时候Cayley-Salmon线是共轭的？

When are the Cayley-Salmon lines conjugate?

给定二次曲线上的六个点，帕斯卡定理产生了一个著名的配置，称为\emph{hexagrammum mysticum}。它包括，除其他事项外，20施泰纳点和20凯莱鲑鱼线。这是一个经典的定理，由于冯施陶德，施泰纳点落入十个共轭对参考圆锥曲线，但这是不正确的C-S线的一般选择六个点。本文证明了当原六元组为~ \emph{tri-involutive}时，C-S线是精确共轭的。三对合六元组的簇是余维数为2的算术Cohen-Macaulay。我们确定了它的$SL_2$ -等变最小分辨。

Rational normal curves and Hadamard products

Given $r>n$ general hyperplanes in $\mathbb P^n,$ a star configuration of points is the set of all the $n$-wise intersection of them. We introduce {\it contact star configurations}, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in $\mathbb P^2$, we prove that the union of two contact star configurations has a special $h$-vector and, in some cases, this is a co…

关于直线的一个Poncelet定理

A Poncelet theorem for lines

我们的目的是证明一个庞斯莱型定理的线配置的复射影。更准确地说，我们说一个多边形有2n条边连接2n个顶点A1，A2，.，A2n在n条线的构形Ln中内接良好，如果构形的每条线恰好包含A1，A2，.，A2n。证明了：设Ln是复射影平面上的n条直线的构形，D是复射影平面上的光滑二次曲线。如果它存在一个多边形与2n边很好地内接在Ln和外切周围D然后有无限多个这样的多边形。特别地，Ln中的一般点是这样的多边形的顶点。“我们提出了一个基于Frégier对合的初等证明。我们首先回顾有关这些对合的一些事实。然后我们探讨以下问题：什么时候对合的乘积对应于对合？它导致帕斯卡定理，其对偶版本证明的Brianchon，并证明其推广莫比乌斯。

Equations for point configurations to lie on a rational normal curve

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to …

$\mathbb P^2$中胖点子格式的Hilbert函数分类

Classifying Hilbert functions of fat point subschemes in $\mathbb P^2$

最近的一份文件的第一和第三作者连同Sabourin提出的问题是什么可能的希尔伯特函数是脂肪点子计划的形式$2p_1+...+2p_r$，所有可能的选择$2p_1+...+2p_r$不同的点在投影平面。本文研究了任意特征的代数闭域$k$上平面上的$r$点的问题，即当$r \le 8$或$r$点位于（可能可约的）二次曲线上时，$r$点的存在性.在这两种情况下，它遵循的工作，第二作者，只有1000多个配置类型的点，其中我们的概念的配置类型是一个推广的概念表示组合几何，也被称为一个表示简单拟阵。(We如果对于非负整数$m_i$、$Z=m_1p_1+...+m_rp_r$和$Z'=m_1p'_1+...+m_rp'_r$的所有选择具有相同的希尔伯特函数，则假定$p_1,...,p_r$和$p'_1,...,p'_r$具有相同的{ \it配置类型}。假设无论是$7 \le r\le 8$（见最近的工作瓜尔多和第二作者的情况$r\le 6$）或点$p_i$谎言的圆锥曲线，我们明确确定所有的配置类型，并显示如何配置类型和系数$m_i$确定（在一个明确的可计算的方式）希尔伯特函数（有时分级贝蒂数）的$Z=m_1p_1+...+m_rp_r$。我们证明了我们的结果，明确列出所有的希尔伯特函数的计划$r\le 8$双点，并为每个希尔伯特函数，我们的状态，精确地点必须如何安排（在配置类型方面），以获得希尔…

Configurations of points and topology of real line arrangements

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties. Using this dual point of view, we construct several examples of co…