Michela Artebani, Igor Dolgachev
This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration.
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This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three.
We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$ there exists a curve $H$ (called the Hesse curve of the pencil) of degree $6(d-1)$ and genus $3(4d^2-13d+8)+1$, and such that: $(i)$ $H$ has $d^2$ singular points of multiplicity three at the base points of the pencil and $3(d-1)^2$ ordinary nodes at the singular points of the degenerate members of the pencil; $(ii)$ for each member of the pencil the intersection of $H$ with this fibre consists of the inflection points of this member and the base points of the pencil.
The Hesse pencil of plane curves and osculating conics
本文利用射影微分几何工具研究黑塞束平面三次曲线的密切圆锥曲线与六点密切点，给出显式坐标公式，并据此构造出新的自由与近自由曲线族。
The Hesse Pencil Variety
研究Hesse pencil变种H_8的结构，证明其维数为8，并通过Schur模块与SL(3)轨道的关系确定其几何性质。
Elliptic-elliptic surfaces and the Hesse pencil
通过 Hesse 纤维束构造 $p_g=q=1$ 椭圆曲面，识别高阶 Noether-Lefschetz locus 分支，并利用 infinitesimal Torelli theorem 及 Shioda modular surface 给出周期映射支配性的多种证明。
In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family …
The Mordell Weil Groups of Cubic Pencils
该研究聚焦三次束基点对Mordell-Weil群的影响，完成了一般位置下含6至8个基点的三次束的探究与分类，并给出了相关应用。
Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves
本文在特征非三的任意域上构造Hesse椭圆曲线的三次同源，对应三次算术几何平均的一步，并应用于特征三中普通椭圆曲线的Serre-Tate典范提升，同时探讨与超几何函数的联系。
Properties of Hesse derivatives of cubic curves
本文研究三次曲线的 Hesse 导数性质，证明了切点与极圆锥曲线的几何关系，并揭示了 Hesse 算子构成迭代离散动力系统，分析了其不动点及闭合轨道。
Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic a topological type (cubic, points), and combinatorial pencil the cyclic sequence of five combinatorial reducible cubics. Up to the action of the symmetric group $S_5$ on $\{2, ...6\}$, there are seven possible combinatorial pencils with node at 1. Consider now the set of six pencils obtained, making the node to be $1, ...6$. Up to the action of S_6 on ${1, …
To a generic configuration of eight points in convex position in the plane, we associate a list consisting of the following information: for all of the 56 conics determined by five of the points, we specify the position of each remaining point, inside or outside. We prove that the number of possible lists, up to the action of $D_8$ on the set of points, is 47, and we give two possible ways of encoding these lists. A generic compex pencil of cubics has twelve singular (nodal) cubics and nine distinct base points, any eight of them determine the ninth one, hence the pencil. If the base points ar…
A real algebraic plane curve $A$ is said to be dividing if its real part $\mathbb{R}A$ disconnects its complex part $\mathbb{C}A$. A pencil of curves is totally real with respect to $A$ if it has only real intersections with $\mathbb{C}A$. If there exists such a pencil, then $A$ is dividing, this is the case for the $M$-curves. Can conversely any dividing curve be endowed with a totally real pencil? We study here the case of $M-2$-sextics having 2 or 6 empty exterior ovals. Such sextics are always dividing. We prove that they may actually be endowed with a totally real pencil of cubics.
The dynamics of the Hesse derivative on the $j$-invariant
研究立方曲线在$j$-不变量集上的Hesse导数动力学，分析其轨道数量及立方曲线与$n$-fold Hesse导数同构的条件，探讨$j$-不变量周期性对曲线周期性的影响。
Pencils of plane cubics with one base point
该研究针对仅一个基点且一般成员光滑的平面三次曲线束展开分类，在不可约假设下证明射影变换下存在唯一非平凡束，并与经典方法做对比。
We present a solvable two-dimensional piecewise linear chaotic map which arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the nontrivial ultradiscrete limit of the solution in spite of a problem known as "the minus-sign problem."
We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they have an associated K3 surface. Thus a pencil gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces, and the baselocus of the original pencil - the intersection of two cubic fourfolds. They both have crepant resolutions which are derived equivalent.
We give two examples of plane curve arrangements of pencil type which are very close to line arrangements, though the action of the monodromy operator on the first cohomology group of the Milnor Fiber has eigenvalues of order 5 and 6, showing that surprising situations can occur for larger classes of curve arrangements than for line arrangements. Our computations rely on the algorithm given by A. Dimca and G. Sticlaru which detects the non trivial monodromy eigenspaces of free curves.
Twelve common flex lines in a general pencil of cubics
本文证明复数域上一般平面三次曲线束有12条公共挠线，回答了Ciliberto等人的问题，其主要结果借助ChatGPT 5.5 Pro和Danus系统这类生成式AI获得。
On the syntomic regulator of the Hesse cubic curves and $p$-adic hypergeometric functions
本文引入新型$p$-adic超几何函数，证明其满足Dwork型同余关系，并用于表达Hesse三次曲线的syntomic regulator，同时建立了与Asakura对数型$p$-adic超几何函数的变换公式。
Geometry of Prym semicanonical pencils and an application to cubic threefolds
本研究探讨模空间中半典范线束对应除子上的Prym映射，揭示其差异与低亏格几何特性，并通过对特定除子的分析得到三次三维簇直线的计数结果。
The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties. A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its $\SL(3)$-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified…
It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.
We present a simple heuristic method to derive the Painlevé differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.
In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in $\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$ of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in $\mathrm{PG}(3,q)$. In particular we determine the point orbit distributions and plane orbit distributions of all $G$-orbits of lines which are contained in an osculating plane of $\mathcal{C}$, have non-empty intersection w…
Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves
本文研究了具有p_g=K^2=1的曲面中曲率2 pencil的存在性，证明其在模空间中形成不可约子流形，并探讨了与平面三次曲线相切的圆锥线几何关系。
For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.
Let $E$ be a smooth elliptic curve over $\mathbb{C}$. For $E$ embedded into $\mathbb{P}^2$ as Hesse cubic and $V$ an Ulrich bundle on $E$ we derive a explicit presentation of $V$ using Moore matrices and theta functions.
Negative Curves and Elliptic Fibrations on a Special Rational Surface
本文针对双重 Hesse 构型吹胀有理曲面上的 $(-1)$-曲线，通过构造修正的 Cremona 分岔图并借助椭圆束闭公式，给出了曲线次数的闭式表达，同时刻画了奇点与方程。
Geometric Endomorphisms of the Hesse moduli space of elliptic curves
本文研究椭圆曲线Hesse模空间上Cayleyan映射与Hessian映射生成的自由半群，分析其几何与动力学性质，并揭示两者在实周期点数量和Julia集上的显著差异。
Intrinsic enumerative mirror symmetry: Takahashi's log mirror symmetry for $(\mathbb{P}^2,E)$ revisited
本文重新审视Takahashi对（P²,E）的对数镜像对称性猜想，通过本质镜像构造和周期积分计算，证明该猜想是本质镜像对称性的自然结果，并推广至任意维数的log Calabi-Yau变种。
Second moments and the bias conjecture for the family of cubic pencils
本文研究1参数椭圆曲线族的二阶矩和，给出三次束族二阶矩的显式公式（通过亏格2曲线点数计数表示），并证明偏差猜想对该三次束族成立。
On the Alexander polynomials of conic-line arrangements
本文计算了与束相关的圆锥曲线-直线排列的Alexander多项式，证明了Halphen束的一般性结果，并构造了具有7次单位根的非约化排列，填补了该类多项式的空白。
We study the reciprocal position of nine points in the plane, according to their collinearities. In particular, we consider the case in which the nine points are contained in an irreducible cubic curve and we give their classification. If we consider two configurations different when the associated incidence structures are not isomorphic, we see that there are 131 configurations that can be realized in $\mathbb{P}^2_{\mathbb{Q}}$, and there are two more in $\mathbb{P}^2_K$, where $K =\mathbb{Q}[\sqrt{-3}]$ (one of the two is the Hesse configuration given by the nine inflection points of a cubi…
Quatroids and Rational Plane Cubics
本文研究非一般8点构型下的有理三次曲线，提出高阶拟阵概念quatroids对构型空间进行分层，计算了全部779777个quatroids，并揭示了各分层的曲线数量及三次曲线束基轨迹等不变量。
On the GIT stability of linear systems of hypersurfaces in projective space
该研究借助几何不变量理论（GIT）解决射影空间中固定次数超曲面线性系统的射影等价分类问题，给出显式判据，并应用于平面三次曲线束、任意指数Halphen束的GIT稳定性分析。
We compute the class of the locus in M_g of curves having a pencil with two unspecified triple ramification points. This is the first example of a geometric divisor on M_g which is not the pull-back of a divisor on the moduli space of pseudo-stable curves. This space, in which elliptic tails are replaced by cusps, appears as a result of the first divisorial contraction in the minimal model program for M_g. In particular, we show that our divisor picks-up the locus of Fermat cubic tails when restricted to the boundary divisor of elliptic tails. We also give various enumerative applications conc…
We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an algebraic set of lines, but it turns into such a set by adding just one pencil of lines.
Computing A1-Euler numbers with Macaulay2
本文利用 Macaulay2 在 GW(k) 中进行富集计数，重新证明三次曲面直线数为 3+12h，并计算多种几何计数及 A1-Milnor 数。
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 $…
Rational elliptic surfaces with six singular double fibres
本文研究具有六个奇异双纤维的有理椭圆面，通过多种视角进行分类并给出构造方法。
The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are actually the same: (A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to P^1 induced by a theta-characteristic; and several more. The corr…
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