Studies on Algebraic Geometry and Coding Theory

代数几何与编码理论研究

• 批准号: 10640006
• 负责人: KATSURA Toshiyuki
• 金额: $ 2.18万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640006/
• 关键词: code; Hamming code; K3 purface; modulic space; formal Brauer group; Cartier operator; Chem class; Artin invariant; カルチェ作用素; ピカール数; チャウ群; アーベル曲面; 正標数; 高さ; フロベニウス写像; K^3曲面; 形式的ブラウアー群; ハイト; モジュライ

As for coding theory, we defined the notion of isolated radius of a code and showed that the isolated radius for the [7, 4, 3]-Hamming code is equal to 2√<2>/7 11.As for algebraic geometry, we mainly investigate the structure of the moduli spaces of K3 surfaces. Let X be a K3 surface defined over an algebraically closed field k of characteristic p>0, and Φ_X the formal Brauer group of X.We denote by h the height of Φ_X, and denote by C the Cartier operator on Z_1, where Z_1 is the sheaf of d-closed 1-forms. We define inductively the sheaf Z_i as Ker dC^<i-1>. Let M be the moduli stack of polarized K3 surfaces of degree 2d (d is a positive integer, d×2d) and let π : χ→M be the universal family. We set υ=π_*Ω^2_<χ/M>. This gives an element of Chow ring of M.For an integer h (1【less than or equal】h【less than or equal】10), we set M^<(h)>={X∈M|height Φ_X【greater than or equal】h}. Let X be a K3 surface with polarization D of degree 2d and let x∈M be a point which corresponds to (X, D), and I … More m H^1(X,Z_h) the image of the homomorphism H^1(X,Z_h)→H^1(X,Ω^1_X) which is induced by the natural inclusion Z_h→Ω^1_X. Assume the height of the formal Brauer group Φ_X is equal to h<∝. Then, we could prove Im H^1(X,Z_h)=21-h, and that the tangent space of M^<(h)> at x is naturally isomorphic to Im H^1(X,Z_h)∩D^⊥⊂H^1(X,Ω^1_X). In particular, we have dim M^<(h)>=20-h. Moreover, we could show that the class of M^<(h)> in the Chow group CH^<h-1>_Q(M) is given by (p^<h-1>-1) (p^<h-2>-1)...(p-1)υ.Now we consider the case of B_2=ρ. For this case, we got some results on the structure of the locus with Artin invariant σ【less than or equal】3 related to the results by Shafarevich, but we don't reach the final understanding. To calculate Chern class of the locus whose Artin invariant is less than or equal to σ is also an interesting problem. We are now investigating this locus, using the method by Pragacz. Similar results hold for abelian surfaces. We are now going to examine the cases of Calabi-Yau manifolds and hypersurfaces. Less

在编码理论方面，我们定义了码的孤立半径的概念，并证明了[7，4，3]-Hamming码的孤立半径等于2√&lt；2&gt；/711。在代数几何方面，我们主要研究了K3曲面模空间的结构。设X是定义在特征为p&gt；0的代数闭域k上的K3曲面，Φ_X是X的形式Brauer群.我们用h表示Φ_X的高度，用C表示Z_1上的卡地亚算子，其中Z_1是d-闭1-型的层.我们归纳地定义层Z_i为Ker dc^&lt；i-1&gt；设M是2d次极化K3曲面的模叠叠(d是正整数，d×2d)，π:χ→M是普适族。我们设置了υ=π_*Ω^2_&lt；χ/M&gt；。对于整数h(1[小于或等于]h[小于或等于]10)，我们设置M^&lt；(H)&gt；={X∈M|HeightΦ_X[大于或等于]h}。设X是偏振度D为2d的K3曲面，x∈M是对应于(X，D)的点，I…设m H^1(X，Z_h)是由自然包含Z_h→^1_X诱导的同态H^1(X，Z_h)ΩH^1(X，→Ω^1_X)的像，假定形式Φ_X的高度等于h&lt；∝。然后，我们可以证明Im H^1(X，Z_h)=21-h，并且M^&lt；(H)&gt；在x处的切空间自然同构于Im H^1(X，Z_h)∩D^⊥⊂H^1(X，Ω^1_X)。特别地，我们有dim M^&lt；(H)&gt；=20-h。此外，我们可以证明Chow群CH^&lt；h-1&gt；_q(M)中的M^&lt；(H)&gt；由(p^&lt；h-1&gt；-1)(p^&lt；h-2&gt；-1)υ给出。现在我们考虑B_2=ρ的情形。对于这种情况，我们得到了与Shafarevich的结果有关的Artin不变量σ[小于或等于]3的轨迹结构的一些结果，但我们没有达成最终的理解。计算Artin不变量小于或等于σ的轨迹的Chern类也是一个有趣的问题。我们现在正在使用Pragacz的方法来研究这个轨迹。类似的结果也适用于阿贝尔表面。我们现在要研究Calabi-Yau流形和超曲面的情况。较少

项目成果

T.Terasoma: "Hodge structure of Gel'fand-Kapranov-Zelsvinski hypergeometric integral and Tursted Ehrhard polynomial"Proc.of the Taniguchi Symposium. 453-476 (1998)

T.Terasoma：“Gelfand-Kapranov-Zelsvinski 超几何积分和 Tursted Ehrhard 多项式的 Hodge 结构”Proc. of the Taniguchi Symposium。

T.KATSURA: "On the distribution of linear codes areumd the [7,4,3]-Hamming code" Max-Plank-Institut fur Mathematik, Prepsint Series. 3. 1-8 (1999)

T.KATSURA：“线性码的分布是 [7,4,3]-Hamming 码”Max-Plank-Institut 数学，Prepsint 系列。

G. van der Geer and T, Katsura: "On a stratification of the moduli of K3 surfaces"J. Eur. Math. Soc.. 2. 259-290 (2000)

G. van der Geer 和 T, Katsura：“关于 K3 表面模量的分层”J。

T.Katsura: "Formal Braver groups and the modulispaces of K3 and Aleutian smtaces in positive characteristic"Proc.of Confi on Algefiaic Geometry, Number Theory, Coding Theory and Cryptyniphy. 107-112 (2001)

T.Katsura：“正式的 Braver 群以及 K3 和 Aleutian smtace 的正特征模空间”Proc.of Confi on Algefiaic Geometry、Number Theory、Coding Theory and Cryptyniphy。

T.Katsura: "On the distribution of linear codes around the [7,4,3]-Hamming code"Max-Planck-Institut fur Mathematik Preprint Series. 99-3. (1999)

T.Katsura：“关于 [7,4,3]-汉明码周围线性码的分布”马克斯普朗克研究所数学预印本系列。