On Differential equation of invariants of binary forms
Leonid Bedratyuk
本文给出了二元形式不变量的一阶偏微分方程的显式表达式，通过求解方程，计算了$n\leq 6$和$n=8.$两种情形下不变量环及其合轨的最小生成集
An explicit form of first order PDE for invariants of binary form are found. By solving the equation a minimal generation set for a ring of invariants and theirs syzygies are calculated in the cases $n\leq 6$ and $n=8.$
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A short explanation of the complexity of the structure of Demailly-semple invariant jet differentials was given
Basic invariants of binary forms over $\mathbb C$ up to degree 6 (and lower degrees) were constructed by Clebsch and Bolza in the 19-th century using complicated symbolic calculations. Igusa extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. In this paper a simple proof is supplied that works in characteristic $p > 5$ and uses some concepts of invariant theory developed by Hilbert (in characteristic 0) and Mumford, Haboush et al. in positive characteristic. Further the analogue for pairs of binary cubics is also treated.
In this article we briefly discuss the finite generation of fiber rings of invariant k-jets of holomorphic curves in a complex projective manifold, using differential Galois theory.
On Hermite's invariant for binary quintics
Hermite不变量H是二元五次对合超曲面的定义方程。本文分析了H.我们确定了这个超曲面的奇异轨迹，并证明它是五次线性协变的完全交集。这个超曲面的投射对偶可以通过对合来与它自身等同。证明了H的Jacobi理想是高度为2的完全理想，并刻画了它的SL_2-等变最小分解。最后一节给出了二元形式协变的向矢的一般形式，并用它计算了H的向矢。
In this paper we describe an algorithm based on the Picard-Vessiot theory that constructs, given any curve invariant under a finite linear algebraic group over the complex numbers, an ordinary linear differential equation whose Schwarz map parametrizes it.
We consider the algebra of invariants of binary forms of degree 10 with complex coefficients, construct a system of parameters with degrees 2, 4, 6, 6, 8, 9, 10, 14 and find the 106 basic invariants.
We consider the algebra of invariants of binary forms of degree 9 with complex coefficients, find the 92 basic invariants, give an explicit system of parameters and show the existence of four more systems of parameters with different sets of degrees.
On complete system of invariants for the binary form of degree 7
计算了7次二元形式的不变代数的齐次生成元的极小系。
We investigate the topology of the closure in a wonderful compactification of the set of unipotent-invariant bilinear forms.
Tropical invariants for binary quintics and reduction types of Picard curves
本文利用二元五次型的热带不变量刻画皮卡曲线的约化类型，并提出适用于任意簇上群作用的热带不变量一般框架，将二元形式问题映射到 $\overline{M}_{0,n}$ 的对称化版本中处理。
We consider the degrees of the elements of a homogeneous system of parameters for the ring of invariants of a binary form, give a divisibility condition, and a complete classification for forms of degree at most 8.
We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way of extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.
We determine the Waring ranks of all sextic binary forms with complex coefficients using a Geometric Invariant Theory approach. Using the five basic invariants for sextic binary forms, our results give a rapid method to determine the Waring rank of any given sextic binary form. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3. We show that for binary forms of arbitrary degree the cactus rank, a.k.a. scheme rank, is determined by the corresponding Waring rank. Finally we determine the border ranks of all…
Differential forms and invariants of complex manifolds
本文综述了由微分形式得到的紧复流形代数不变量，涵盖上同调群、陈类、有理同伦群和高阶运算，并总结了相关结果与未解决问题。
We provide a geometric characterisation of binary sextics with vanishing quadratic invariant.
Let $\mathcal{I}_{d_1,d_2}$ and $\mathcal{C}_{d_1,d_2}$ be the algebras of joint invariants and joint covariants of the two binary forms of degrees $d_1$ and $d_2.$ Formulas for computation of the Poincaré series $\mathcal{PI}_{d_1,d_2}(z),$ $ \mathcal{PC}_{d_1,d_2}(z)$ of the algebras is found. By using these formulas, we have computed the series for $d_1,d_2 \leq 20.$
Relative period realization of holomorphic differentials with prescribed invariants
本文完整描述了黎曼曲面上具有给定零点阶数和超椭圆结构、自旋奇偶性等不变量的全纯微分形式的可实现相对周期表示，回答了Simion Filip的问题。
I show that the cohomology of the generic points of algebraic complex varieties becomes {\sl stable} birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'.
Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal h…
A Computational Solution to a Question by Beauville on the Invariants of the Binary Quintic
我们得到了另一个证明的内射性结果Beauville的地图从模空间的四次del Pezzo表面的一组共轭类的某些子群的Cremona组。这相当于表明，一个投影配置的五个不同的无序点的线可以重建从它的五个投影四点子配置。这是通过减少到一个问题的经典不变理论的二进制五次，这是解决了计算机辅助的方法。更确切地说，我们表明，6个特定的不变量的程度24，建设的解释给我们的Beauville，产生所有不变量的程度，其中是可分的48。
Classifying binary quadratic forms using Clifford invariants
本文利用Clifford不变量对二元二次型进行分类，并推广高斯组合与二次代数Picard群的关系。
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