Enumeration of rational plane curves tangent to a smooth cubic

On the enumeration of rational plane curves with tangency conditions

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass through 3d-1-a-2b-c general points of P^2? The answer is given as a generalization of Kontsevich's recursion. We also investigate more general enumerative problems of this sort, and prove an analogue of a formula of Caporaso and Harris.

Rational points of elliptic surfaces and Zariski $N$-ples for cubic-line, cubic-conic-line arrangements

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.

光滑四次三维簇自同构群的格论方法

Automorphism Groups of Smooth Cubic Fourfolds through Lattice Theory

本文利用格论和OSCAR枚举算法，完成了42族光滑四次三维簇的所有可能自同构群分类，基于周期映射已知结果。

完全交的网对数切层

Net logarithmic tangent sheaves of complete intersections

本文定义了网对数切层作为对数切层的推广，适用于光滑完全交簇及其子簇，并研究了光滑三次曲面上此类切层在模空间中的轨迹。

曲率2的曲面上的pencil，包络线以及与平面三次曲线相切的圆锥线

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。

Enumeration of singular varieties with tangency conditions

We construct the algebraic cobordism theory of bundles and divisors on smooth varieties. It has a simple basis (over Q) from projective spaces and its rank is equal to the number of Chern invariants. As an application we study the number of singular subvarieties satisfying given tangent conditions with a fixed smooth divisor, where the subvariety is the zero locus of a section of a vector bundle. We prove that the generating series gives a homomorphism from the algebraic cobordism group to the power series ring. This implies that the number of singular subvarieties with tangency conditions is …

A new model for the theta divisor of the cubic threefold

In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold $X$. We use the standard realization of $X$ as a conic bundle and a $4-$dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle structure. The additional data of an even theta characteristic on the curves in the family gives us a model for the theta divisor.

On the number of Mordell-Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contain…

Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. In this paper we present numerical data regarding the analogous statement for cubic surfaces. For the surfaces examined, we also test Manin's conjecture relating the asymptotics of rational points of bounded height on a Fano variety with the rank of the Pi…

The enumerative geometry of cubic hypersurfaces: point and line conditions

In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Comput…

Neuberg cubics over finite fields

The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides interesting potential invariants for elliptic curves, focussing on an explicit example over $\mathbb{F}_{23}$. We also prove that tangent conics for a Weierstrass cubic are identical or disjoint.

Constructing Cubic Curves with Involutions

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions.

Cubic fourfolds containing a plane and a quintic del Pezzo surface

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffi…

非退化 Hermitian 簇与三次超曲面的最大交点数

Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface

本文证明了 Edoukou-Ling-Xing 猜想在三次超曲面情形（d=3，q≥7）下成立，即非退化 Hermitian 簇与三次超曲面的最大交点数仅由 d 个特殊超平面并集达到。

通过 WDVV 方程对有理尖点曲线的枚举

Enumeration of Rational Cuspidal Curves via the WDVV equation

本文扩展 Kontsevich 递归公式思想，提出射影平面有理尖点曲线特征数的猜想公式，并基于此技术扩展计算具有 E6 奇点的有理四次曲线特征数。

有限域$\u211d_2$上的三次四折形普查

A census of cubic fourfolds over $\mathbb{F}_2$

本研究完成了有限域$\u211d_2$上三次四折形的完整同构类计算，统计了其各类不变量，改进了轨道枚举方法以提升计算效率，并关联了K3曲面zeta函数实现问题与判别式补渐近框架。

Counting singular curves with tangencies

We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. Combined with the results of S. Basu and R. Mukherjee, this gives us a complete formula for the characteristic number of curves with $δ$-nodes and one singularity of type $A_k$, tangent to a given line, provided $δ+ k \leq 8$. We use a topological method, namely the method of dynamic intersections to compute the degenerate…

通过 WDVV 研究具有一阶切触的热带有理曲线

Tropical rational curves with first order tangency via WDVV

本文利用 WDVV 技术的热带类比，计算了 $\mathbb{CP}^2$ 中具有一阶切触的热带有理曲线计数，验证了结果与复几何中的早期结果一致。

About the group law for the Jacobi variety of a hyperelliptic curve

We generalize the group law of curves of degree three by chords and tangents to the Jacobi variety of a hyperelliptic curve. In the case of genus 2 we accomplish the construction by a cubic parabola. We derive explicit rational formulas for the addition on a dense set in the Jacobian.

Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^…

关于Caporaso-Harris公式的纤维丛版本

On a fibre bundle version of the Caporaso-Harris formula

本文推广Caporaso-Harris公式，提出纤维丛版本，并解决P^3中平面节点三次曲线特征数计数问题。

A Mordell-Weil theorem for cubic hypersurfaces of high dimension

Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on…

三次曲线的 Hesse 导数的性质

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。

A table of elliptic curves over the cubic field of discriminant -23

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics ("old a…

双投影曲面的Bourbaki次数

Bourbaki degree of pairs of projective surfaces

本文研究双同次多项式序列的对数切sheaf，引入两个正整数不变量，分析极小syzygies次数与不变量的关系，通过叶状结构理论工具，分类了立方族和二次立方对，反驳了2018年Calvo-Andrade等人的猜想。

Real Space Sextics and their Tritangents

The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on space sextics that arise from del Pezzo surfaces of degree one. Their numbers of planes that are tangent at three real points vary widely; both 0 and 120 are attained. This solves a problem suggested by Arnold Emch in 1928.

Bigness of the tangent bundle of del Pezzo surfaces and $D$-simplicity

We consider the question of simplicity of a ring $R$ under the action of its ring of differential operators $D_R$. We give examples to show that even when $R$ is Gorenstein and has rational singularities $R$ need not be a simple $D_R$-module; for example, this is the case when $R$ is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when $R$ is the homogeneous coordinate ring of…

关于$\u200b\mathrm{PG}(3,q)$中扭三次曲线外部线类$\u200b\mathcal{O}_6$的轨道与关联矩阵的进一步结果

Further results on orbits and incidence matrices for the class $\mathcal{O}_6$ of lines external to the twisted cubic in $\mathrm{PG}(3,q)$

该研究针对射影空间$\u200b\mathrm{PG}(3,q)$中研究最少的扭三次曲线外部线类$\u200b\mathcal{O}_6$，提出其新轨道族，分析对应稳定子群与关联子矩阵，通过求解相关三次、四次方程解数完成推导。

尖点三次射影叶状结构：整体刚性、形变与拉回

Cuspidal Cubic Projective Foliations: Global Rigidity, Deformations, and Pull-Backs

本文研究不可约尖点三次曲线相关的全纯叶状结构，证明次数至多七的整体刚性分类，引入尖点线性拉回轨迹并证明其刚性，主要贡献在于分类与拉回识别。

具有少量二重点的单纯排列

Simplicial arrangements with few double points

本文通过几何论证证明，在单纯排列中相关三次曲线必可约，从而在二重点线性界限下证实了Grünbaum的渐近分类猜想。

On projective varieties $n$-covered by curves of degree $δ$

As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $δ\geq n-1$ we have $N\leq π(r,n,δ+r(n-1)+2)$, where $π(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ of degree $d$. A lot of examples of varieties as in the title and attaining the previous bound for the embedding dimension are constructed from Castelnuovo varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is also…

Obstructions to deforming space curves and non-reduced components of the Hilbert scheme

Let $Hilb P^3$ denote the Hilbert scheme of smooth connected curves in $P^3$. We consider maximal irreducible closed subsets $W \subset Hilb P^3$ whose general member $C$ is contained in a smooth cubic surface and investigate the conditions for $W$ to be a component of $(Hilb P^3)_{red}$. We especially study the case where the dimension of the tangent space of $Hilb P^3$ at $[C]$ is greater than $\dim W$ by one. We compute obstructions to deforming $C$ in $P^3$ and prove that for every $W$ in this case, $Hilb P^3$ is non-reduced along $W$ and $W$ is a component of $(Hilb P^3)_{red}$.

On the multiplicity of reducible relative stable morphisms

Let $(Z, D)$ be a pair of a smooth surface and a smooth anti-canonical divisor. Denote by $\mathfrak{M}_β$ the moduli stack of genus $0$ relative stable morphisms of class $β$ with full tangency to the boundary. Let $C_1$ and $C_2$ be rational curves fully tangent to $D$ at the same point $P$ and assume that $C_1$ and $C_2$ are immersed and that $(C_1.C_2)_P=\min\{D.C_1, D.C_2\}$. Then we show that the contribution of $C_1\cup C_2$ to the virtual count of $\mathfrak{M}_{[C_1]+[C_2]}$ is $\min\{D.C_1, D.C_2\}$. As an example, we describe genus $0$ relative stable morphisms to $(\mathbb{P}^2, (\…

Geometry of Smooth Extremal Surfaces

We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces $\textit{extremal surfaces}$, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface $X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which…

Unbendable rational curves of Goursat type and Cartan type

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of thei…

Hurwitz Theory of Elliptic Orbifolds, I

An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for the full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to the enumeration of branched covers of a pillowcase, i.e. the quotient of an elliptic curve by the elliptic involution, proving quasi-modularity for $Γ_1(2)$. We generalize their work to the quotient of an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving quasi-m…