| Co-Investigator(Kenkyū-buntansha) |
KATO Akishi Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (10211848)
KOHNO Toshitake Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (80144111)
KAWAMATA Yujira Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90126037)
TERADA Itam Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (70180081)
TERASOMA Tomohide Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50192654)
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| Research Abstract |
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
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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
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