Mark Bly
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being selected.
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This is a note constructing a certain weight 4 automorphic form on the moduli space of cubic surfaces, posted here because it is referred to in math.AG/0002066
We shall show the existence of 15 automorphic forms of weight 8 on the moduli space of marked Hessian quartic surfaces of cubic surfaces. These automorphic forms can be interpreted in terms of the coefficients of the Sylvester form of a general cubic surface.
The moduli space of cubic surfaces in complex projective space is known to be isomorphic to the quotient of the complex 4-ball by a certain arithmetic group. We apply Borcherds' techniques to construct automorphic forms for this group and show that these provide an embedding of the moduli space in 9-dimensional projective space. We also show that our automorphic forms directly encode the geometry of cubic surfaces, by showing that each of Cayley's invariants (certain cross-ratios) is simply a quotient of two of our automorphic forms.
We shall construct a 5-dimensional linear system of holomorphic automorphic forms on a 3-dimensional complex ball by applying Borcherds theory of automorphic forms. We shall show that this linear system gives the dual map from the Segre cubic 3-fold to the Igusa quartic 3-fold.
We identify the algebra of regular functions on the space of quartic polynomials in three complex variables invariant under SL(3,C) with an algebra of meromorphic automorphic forms on the complex 6-ball. We also discuss the underlying geometry.
Local Constancy of the Betti Automorphic Category over the Moduli of Curves
本文采用纯微局部方法证明了曲线模空间上Betti自守范畴的局部常性猜想，该结果不依赖新近确立的Betti几何朗兰兹猜想，还验证了Ben-Zvi与Nadler关于Betti自守范畴仅依赖曲线亏格的猜想。
Automorphic gluing functor in Betti Geometric Langlands
本文研究曲线结点退化中的幂零层自同构范畴，通过标记点鼓出射影直线解释仿射Hecke算子，并构造连接结点曲线与光滑化自同构范畴的粘合函子。
This paper is the written version of D.Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and non-archimedian). The talk is based on joint works of D.Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.
We introduce interaction entropies, which can be represented as logarithmic couplings of certain cycles on a class of algebraic curves of arithmetic interest. In particular, via interaction entropies for Legendre-Ramanujan curves $ Y^n=(1-X)^{n-1}X(1-αX)$ ($ n\in\{6,4,3,2\}$), we reformulate the Kontsevich-Zagier integral representations of weight-4 automorphic Green's functions $ G_2^{\mathfrak H/\overline{\varGamma}_0(N)}(z_1,z_2)$ ($N=4\sin^2(π/n )\in\{1,2,3,4\}$), in a geometric context. These geometric entropies allow us to establish algebraic relations between certain weight-4 automorphi…
We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra $\mathcal{H}_{G,C}$. In the case $G=\rm{GL}_2$, we construct a commutative subalgebra in $\mathcal{H}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G=\rm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. I…
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abe…
Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms
本文在p-adic Mumford曲线的K-有理点上构造自伴算子，生成Feller半群，通过热方程谱的极限点对应微分形式零点，从而从谱恢复亏格，超椭圆情形可显式提取。
In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.
An expository description of smooth cubic curves in the real or complex projective plane.
We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic.
We study the sheaves of logarithmic vector fields along smooth cubic curves in the projective plane, and prove a Torelli-type theorem in the sense of Dolgachev-Kapranov for those with non-vanishing j-invariants.
We find all K-polystable smooth Fano threefolds that can be obtained as blowup of projective space along the disjoint union of a twisted cubic curve and a line.
We describe the Hilbert schemes parametrizing curves on a cubic threefold of degree at most 5. In a forthcoming paper, we use this description to give a new proof and extension of a theorem of Iliev, Markushevich and Tikhimirov.
We discuss the GIT moduli of semistable pairs consisting of a cubic curve and a line on the projective plane. We study in some detail this moduli and compare it with another moduli suggested by Alexeev. It is the moduli of pairs (with no specified semi-abelian action) consisting of a cubic curve with at worst nodal singularities and a line which does not pass through singular points of the cubic curve. Meanwhile, we make a comparison between Nakamura's compactification of the moduli of level three elliptic curves and these two moduli spaces.
We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.
Note on smoothness condition on tropical elliptic curves of symmetric truncated cubic forms
本文给出了对称截断三次型系数产生光滑热带曲线的显式条件，并分析了特定细分类型对应的非光滑情形。
Compact Moduli Spaces of Marked Cubic Plane Curves
本文通过几何不变理论研究标记立方平面曲线的紧致模空间，分析奇点和点位置以理解模论中的墙交叉现象。
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.
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first part of the 16th Hilbert problem appear in this classification. One is the distribution of the components of the curve between the components of the non-connected cubic surface which turns out to depend on the patterns of arrangements (see Theorem 1). The other is presence of positive genus among the components of the complement and genus-related restrictio…
We construct and study Sarkisov links obtained by blowing up smooth space curves lying on smooth cubic surfaces. We restrict our attention to the case where the blowup is not weak Fano. Together with the results of arXiv:1106.3716 which cover the weak Fano case, we provide a classification of all such curves. This is achieved by computing all curves which satisfy certain necessary criteria on their multisecant curves and then constructing the Sarkisov link step by step.
Rational curves on cubic hypersurfaces in positive characteristic
研究特征不等于2、3的三次超曲面上有理曲线的模空间，证明当超曲面维度大于等于4时，Kontsevich模空间是不可约的。
We show how the study of the geometry of the nine flex tangents to a cubic produces pseudo-parameterizations, including the ones given by Icart, Kammerer, Lercier, Renault and Farashahi, and infinitely many new ones.
Legendrian curves in $\mathbb C P^3$: cubics and curves on a quadric surface
本文证明了 $\mathbb C P^3$ 中经过三个一般点和一条直线的 Legendrian 有理三次曲线数量为三，并对二次曲面上的所有 Legendrian 曲线进行了分类。
Automorphic functions for square-zero extensions of curves over finite fields
该研究针对有限域上曲线的平方零扩张，推广了PGL₂的自守函数相关结果，证明了PGL₃下某猜想的新案例，还确定了对应球面尖与赫克有限函数的最优支撑界。
In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version modified by Ph. Ellia) concerning maximal families of space curves lying on a smooth cubic surface, assuming the quadratic normality of its general members. We also give a sufficient condition for curves lying on a cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as …
For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, we establish necessary and sufficient conditions for $\mathcal{F}$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane cubics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine explicit formulas for the number $N_q(\mathcal{F})$ of $\mathbb{F}_q$-rational points on $\mathcal{F}$. For the remaining Fermat curves, nice upper bounds for $N_q(\mathcal{F})$ are immediately given by the Stöhr-Voloch Theory.
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.
(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10 and analyse the family of such surfaces and 3-folds in detail when d=6. In particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible ones with non-normal singularities.
Automorphic commutator relations in relative geometric Langlands
本文证明了 Liu-Wang 猜想中的自守换位子关系，通过引入万有-局部叠并利用融合与纯性，移除了此前对曲线亏格和上同调消失的假设，从而对所有光滑射影曲线确立了该几何输入。
In this paper, we study the geometry of two-torsion points of elliptic curves in order to distinguish the embedded topology of reducible plane curves consisting of a smooth cubic and its tangent lines. As a result, we obtain a new family of Zariski N-ples consisting of such curves.
We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the relation between M and the Hilbert scheme Hilb_H(P_3) of cubic space curves. We also compute the Betti numbers of the Chow groups of the moduli space M.
Torsion of elliptic curves over cyclic cubic fields
本研究明确了循环三次域、非循环全实三次域与复三次域上椭圆曲线的全部可能挠群，填补了相关数论研究的空白。
Let C be a smooth cubic curve in the complex projective plane. We show that for every positive integer k, there are only finite number of rational curves of degree k each intersects the cubic C at exactly one point. The number of such rational curves is known when k is 1, as a smooth cubic has 9 flexes points. This number seems to be closely related to the number of plane rational curves of degree k passing through 3k-1 general points, which has been computed.
We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.
This work deals with an algebro-geometric theory of solutions of the classical Yang-Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstrass cubic curves.
In this note we show that the apolar cubic forms associated to codimension two linear sections of canonical curves of genus at least eleven are special with respect to their presentation as sums of cubes.
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.
Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce $n$-contact curves to $E$ and consider its application.
For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves intersecting $B$ at only one point.
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。
Analog of theta-lifting for a curve over dual numbers over a finite field
本文研究了有限域对偶数环上曲线的自守函数框架下的theta提升实例，证明其可由自守函数空间的轨道分解理解，且所有强尖点函数均可通过适当双重覆盖的theta提升构造。
Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.
In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.
We compute the canonical model of the cover of Shimura curves $X_0(2) \to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper "Shimura curves for level 3 subgroups of the (2,3,7) triangle group". Last, we list the coordinates of some rational CM points on X(1).
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_α(T)$ be the upper level sets of Lelong numbers $ν(T,x)$ of $T$. For any $α\geq \frac{1}{3}$, we show that $|V_α(T)\setminus C|\leq 2$ for some cubic curve $C$.
Construction of free curves by adding osculating conics to a given cubic curve
本文从平面三次曲线出发，通过添加其超密切二次曲线构造自由及近自由曲线族，给出结点三次与费马三次曲线的相关结果，并借助群作用揭示费马三次曲线27条超密切二次曲线的几何性质。
We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.
Every smooth cubic plane curve has 9 inflection points, 27 sextatic points, and 72 ``points of type nine". Motivated by these classical algebro-geometric constructions, we study the following topological question: Is it possible to continuously choose $n$ distinct unordered points on each smooth cubic plane curve for a natural number $n$? This question is equivalent to asking if certain fiber bundle admits a continuous section or not. We prove that the answer is no when $n$ is not a multiple of 9. Our result resolves a conjecture of Benson Farb.
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."
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z…
This paper has been rendered obsolete by our newer eprint alg-geom/9411005 "Bott's formula and enumerative geometry", which is a considerably expanded version of the same paper, in spite of the change of titles. Please download alg-geom/9411005 instead, and update possible references.
Cubic points on dynamical modular curves
本文针对与二次多项式映射关联的动力模曲线族，判定含无穷多三次点的曲线类型，据此给出三次域上二次多项式预周期点的分类结论，拓展了相关已有研究成果。
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