Danwakai2004

談話会 (2005年度)

埼玉大学理学部数学教室では不定期に談話会を開催しています。
内容に関心のある方は御参加下さい。

場所：埼玉大学理学部１号館（３階）基礎数理演習室
（お茶会の場所は理学部1号館3階のコモンルームです。）

12月9日(金) 16: 00--17: 00 (15:30からお茶)
宮西正宜氏 (関西学院大学)
Affine pseudo-planes and the Jacobian Conjecture

アブストラクト：
An affine pseudo-plane is one of the satellite surfaces of the affine plane and is defined to be a complex smooth affine surface $X$ of Picard number zero and with no non-constant invertible functions such that there is an ${\mathbf A}^1$-fibration parametrized by the affine line and equipped with a unique multiple fiber of multiplicity, say $d$. Then all the fibers of the ${\mathbf A}^1$-fibration are irreducible, the Picard group of $X$ is isomorphic to ${\mathbf Z}/d{\mathbf Z}$ and it is equal to the fundamental group $\pi_1(X)$. An example is a quotient of a Danielewski surface $xy=z^d-1$ by a ${\mathbf Z}/d{\mathbf Z}$-action given by $\zeta\cdot(x,y,z)=(\zeta x, \zeta^{-1}y, \zeta^a z)$ with $0 < a < d$ and $\gcd(a,d)=1$, where $\zeta$ is a primitive $d$-th root of unity. Another example is the complement in ${\mathbf P}^2$ of a cuspidal rational curve $X_0X_1^{d-1}=X_2^d$. These two examples are slightly different in properties. In terms of the additive group actions, the former has two independent actions, though the latter has only one. An affine pseudo-plane has a good similarity to the affine plane and hence plays a role of a test surface to various problems which are expected to be affirmative for the affine plane and are examplified by the cancellation problem and the Jacobian conjecture. Tom Dieck, generalizing the original construction by Danielewski, produced several counterexamples to the cancellation problem by using the universal coverings of certain affine pseudo-planes which admit algebraic torus actions. We can produce counterexamples to the cancellation problem by using affine pseudo-planes themselves. We shall discuss the Jacobian conjecture for affine pseudo-planes. We yet have no counterexamples but do have some counterexamples if we drop the condition that the ${\mathbf A}^1$-fibration has only one multiple fiber. Furthermore, by widening the class of surfaces, the Jacobian Conjecture will admit more algebro-geometric approaches. We shall report also this algebro-geometric approach.

過去の記録

10月21日(金) 16: 00--17: 00 (15:30からお茶)
阿部孝順氏 (信州大学)
可微分軌道体の微分同相群の１次元ホモロジー群とその応用

9月9日(金) 16: 00--17: 00 (15:30からお茶)
林忠一郎氏 (日本女子大学)
The number of Reidemeister moves for splitting a link

5月27日(金) 16: 00--17: 00 (15:30からお茶)
坂元国望氏 (広島大学)
Front motion in viscous conservation laws with stiff source terms

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