Explicit Study of Algebraic Varieties

(1)通过与Due Tai Pho的合作，我们证明了特征为5的超奇异K3曲面的Artin不变量为λ 3的单合理性. (2)对复正规代数K3曲面上有理双点的所有可能构型进行了分类，并讨论了特征充分大的超奇异K3曲面上有理双点的同样分类. (3)研究了复数域上的一个代数簇通过基域到复数域的各种嵌入而得到的复代数簇的超越格的同构类集，并给出了复数域上的自同构下共轭的非同胚复代数簇的许多例子，特别是研究了奇异的情形.利用虚二次域的类域理论，构造了K3曲面.作为应用，我们得到了奇异K3曲面基场次数的一个下界。(4)假设我们通过定义在数域上的奇异K3曲面X的基域在有限位置P的约化得到超奇异K3曲面X（P）。本文研究了X的Neron-Severi格在X（P）的正交补中的性质，并证明了一个类似于transvendental格的结果。(5)证明了Grassmannian簇中对偶簇的补的拓扑基本群的Lefschetz超平面截口定理，并研究了该基本群与穿孔Riemann曲面的Barid群的Zankiki-van坎彭型关系.

(1) By a joint work with Due Tai Pho, we proved the unirationality of supersingular K3 surfaces in characteristic 5 with Artin invariant 〓 3.(2) We classified all possible configurations of rational double points on complex normal algebraic K3 surfaces, and discussed the same classification for supersingular K3 surfaces in sufficiently large characteristics.(3) We investigated the set of isomorphism classes of transcendental lattices of complex algebraic varieties obained from a single algebraic variety defined over a number field by various embeddings of the base field into the complex number field, and produced many examples of non-homeomorphic complex algebraic varieties that are conjugate under the automorphism of the complex number field.In particular, we investigate the case of singular. K3 surfaces by means of the class field theory of imaginary quadratic fields. As an application, we obtained a lower bound of the degree of the base field of singular K3 surfaces.(4) Suppose that we obtain a supersingular K3 surface X (P) by reduction at a finite place P of a base field of a singular K3 surface X defined over a number field. We investigated the orthogonal complement of the Neron-Severi lattice of X in that of X (P), and proved an analogous result as the case of the transvendental lattices.(5) We proved a Lefschetz hyperplane section theorem for the topological fundamental group of the complement of the duial variety in the Grassmannian variety, and invetigate the Zariski-van Kampen type relation of that fundamental group with the barid group of a punctured Riemann surface.