Studies on Algebraic Geometry and Coding Theory

在编码理论方面，我们定义了码的孤立半径的概念，证明了[7，4，3]-Hamming码的孤立半径等于2 π<2>/7 11。在代数几何方面，我们主要研究了K3曲面的模空间的结构。设X是定义在特征p&gt;0的代数闭域k上的K3曲面，Φ X是X的形式Brauer群，h表示Φ X的高度，C表示Z_1上的Cartier算子，其中Z_1是d-闭1-形式的层。我们归纳地定义层Z_i为Ker dC^<i-1>。设M是2d次极化K3曲面的模堆叠（d是正整数，d×2d），π：χ→M是泛族.我们设置π =π_*Ω^2_&lt;χ/M&gt;。这给出了M的Chow环的一个元素.对于整数h（1[小于或等于]h[小于或等于]10），设M^&lt;（h）&gt;={X∈M|高度Φ_X[大于或等于]h}。设X是K3曲面，其偏振度D为2d，x∈M是对应于（X，D）的点，且I ...更多信息 m H^1（X，Z_h）是同态H^1（X，Z_h）→H^1（X，Ω^1_X）由自然包含Z_h→Ω^1_X导出的象。设形式Brauer群的高度Φ_X等于h&lt;λ。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）H^1（X，Ω^1_X）。特别地，我们有dim M^&lt;（h）&gt;=20-h。此外，我们还可以证明Chow群CH^ _Q（M）中M^&lt;（h）&gt;的类<h-1>由（p^<h-1>-1）（p^<h-2>-1）.给出。(p-1)现在我们考虑B_2=ρ的情况。对于这种情形，我们得到了与Shafarevich的结果相联系的关于Artin不变量σ[小于或等于]3的轨迹的结构的一些结果，但我们没有达到最终的理解。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。我们现在正在研究这个轨迹，使用Pragacz的方法。类似的结果也适用于阿贝尔曲面。我们现在要研究卡-丘流形和超曲面的情况。少

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