On complete system of invariants for the binary form of degree 7
Leonid Bedratyuk
计算了7次二元形式的不变代数的齐次生成元的极小系。
A minimal system of homogeneous generating elements of the invariants algebra for the binary form of degree 7 is calculated.
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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 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…
On complete system of covariants for the binary form of degree 7
计算了7度二进制形式的协变代数的147个齐次生成元的极小系统。
On Differential equation of invariants of binary forms
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
Automorphisms of the ring of invariants of the binary quintic representation of SL2
本文证明SL2二元五次表示的不变环的自同构群等于基域的单位群，并由此推出SL2-等变多项式自同构群同构于单位群。
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.
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。
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.
We investigate the topology of the closure in a wonderful compactification of the set of unipotent-invariant bilinear forms.
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。
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。
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 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 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 $d_1,d_2 \leq 20.$
We offer a Maple package {\tt Poincare\_Series} for calculating the Poincaré series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of Weitzenböck derivations and for the multivariate Poincaré series of algebras of joint invariants/covariants of several binary forms.
On complete system of covariants for the binary form of degree 8
计算了8度二进制形式的协变代数的齐次生成元的极小系统。
We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural $\text{GL}_2$-action on the variables. In particular, we show that these $\text{GL}_2$ orbits are in bijection with pairs of $j$-invariant $0$ elliptic curves together with $3$-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence.
We define one-point disk invariants of a smooth projective Calabi-Yau (CY) complete intersection (CI) in the presence of an anti-holomorphic involution via localization. We show that these invariants are rational numbers and obtain a formula for them which confirms, in particular, a conjecture by Jinzenji-Shimizu.
Using localization methods on the moduli space of stable quasimaps to Grassmannian, we give explicit formulas of $J$-function and double $J$-function of stable quasimaps for complete intersection in Grassmannian.
By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double $J$-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants.
In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of degree $n(d-2)$ by means of assigning every form with nonvanishing discriminant the so-called associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof o…
We study genus 1 Gromov-Witten invariants of Fano complete intersections in the projective spaces. Among other things, we show a reconstruction theorem for genus 1 invariants with only ambient insertions, and compute the genus 1 invariants with 1 marked point. For cubic hypersurfaces of dimension $\neq 4$ and odd dimensional intersections of two quadrics, we obtain a complete reconstruction theorem for genus 1 Gromov-Witten invariants.
We obtain mirror formulas for the genus 1 Gromov-Witten invariants of projective Calabi-Yau complete intersections. We follow the approach previously used for projective hypersurfaces by extending the scope of its algebraic results; there is little change in the geometric aspects. As an application, we check the genus 1 BPS integrality predictions in low degrees for all projective complete intersections of dimensions 3, 4, and 5.
We use the description of the Picard modular surface for discriminant $-3$ as a moduli space of curves of genus $3$ to generate all vector-valued Picard modular forms from bi-covariants for the action of ${GL}_2$ on the space of pairs of binary forms of bidegree $(4,1)$. The universal binary forms of degree $4$ and $1$ correspond to a meromorphic modular form of weight $(4,-2)$ and a holomorphic Eisenstein series of weight $(1,1)$.
The goal of this paper is to explore the genus and degree of the Fano scheme of linear subspaces on a complete intersection in a complex projective space. Firstly, suppose that the expected dimension of the Fano scheme is one, we prove a genus-degree formula. Secondly, we give a degree formula for the Fano scheme.
When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic
In our previous work, we provided an algebraic proof of the Zinger's comparison formula between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension two or three in a projective space. In this paper, we extend the result in any dimensions and for descendant invariants.
Invariant notions of a class of Segre varieties $\Segrem(2)$ of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains $\Segrem(2)$ and is invariant under its projective stabiliser group $\Stab{m}{2}$. By embedding PG(2^m - 1, 2) into \PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant under $\Stab{m}{2}$ as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according…
The ring of $D_4$ triality invariants
该研究证明满足特定尖点条件的 D₄ 三元对合不变量环，同构于二元三次型与二元二次型的联合共变式环。
We compute the alpha invariant of any smooth complex projective spin complete intersection of complex dimension $1 \; ({\rm mod} \; 4)$. We prove that the alpha invariant depends only on the total degree and Pontryagin classes. Our findings are consistent with a long-standing conjecture, often called the Sullivan Conjecture, which states that two complete intersections with the same dimensions, total degrees, Pontryagin and Euler classes are diffeomorphic.
On the Genus-One Gromov-Witten Invariants of Complete Intersections
如前所述，在$¶$中的稳定亏格1全纯映射的模空间的主分量$\ov\M_{1,k}^0(¶,d)$上，全纯向量丛截面的某些自然生成的锥有一个定义良好的Escheriche类。本文将这一结果推广到$J$ -全纯映射的模空间中。我们表明，这种锥的类相关的减少genus-one Gromov-Witten不变量的完整的交叉相应的GW-不变量的环境射影空间。因此，完全交的标准亏格1 GW-不变量可以用射影空间的亏格0和亏格1 GW-不变量来表示。我们明确地陈述了完全相交的三重关系。一个更高的属的不变量的关系也被证明。
We describe the structure of mirror formulas for genus 0 Gromov-Witten invariants of projective complete intersections with any number of marked points and provide an explicit algorithm for obtaining the relevant structure coefficients. The structural description alone suffices for some qualitative applications, such as vanishing results and the bounds on the growth of these invariants predicted by R. Pandharipande.
We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten in…
The ring of Weyl invariant $E_8$ Jacobi forms
研究外尔不变的\(E_8\)弱雅可比形式环，证明其与二元六次式和二元四次式联合协变量环同构且有限生成，借助已知协变量环生成元获极小生成元基。
Let Y be a Calabi-Yau complete intersection in a weighted projective space. We show that the space of quadratic invariants of the hypergeometric group associated with the twisted I-function is one-dimensional, and spanned by the Gram matrix of a split-generator of the derived category of coherent sheaves on Y with respect to the Euler form.
A. Zinger defined reduced Gromov-Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). H. -L. Chang and J. Li provided a proof of the comparison theorem for quintic Calabi-Yau 3-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. T. Coates and C. Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger and proved the comparison theorem for every Calabi-Yau threefold. In this paper,…
Strong u-invariant and Period-Index bound for complete ultrametric fields
本文研究完备超度量域的u-不变量与强u-不变量，通过剩余域描述之，并推广到函数域，同时给出Brauer-l-维数的界。
Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.
Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space
本文研究了加权射影空间中完备交集的椭圆虚拟结构常数及其与格罗莫夫-王特征值的关系。
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