Danielewski area

In mathematics, a Danielewski surface represents a generalization of space and, from the point of view of complex analysis, has properties similar to . ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \mathbb {C} ^{2}}$

definition

A Danielewski surface is an algebraic surface that is algebraically isomorphic to a hypersurface that is defined as the set of zeros of a polynomial , where a polynomial is in one variable. ${\displaystyle S\subset \mathbb {C} ^{3}}$ ${\displaystyle x^{n}\cdot y-p(z)\in \mathbb {C} [x,y,z]}$ ${\displaystyle p\in \mathbb {C} [z]}$

Elementary properties

• In the special case is isomorphic to . ${\displaystyle n=1,\;p(z)=z}$ ${\displaystyle S=\left\{(x,y,z)\in \mathbb {C} ^{3}:x\cdot y=z\right\}}$ ${\displaystyle \mathbb {C} ^{2}}$
• If and only if the polynomial has only simple zeros, it is not only an algebraic surface, but also a complex manifold , since it has no singularities. ${\displaystyle p}$ ${\displaystyle S}$
• Be different with pairs . Then: ${\displaystyle p(z)=(z-a_{1})\cdot (z-a_{2})\cdots (z-a_{m})}$ ${\displaystyle a_{i},a_{j}}$

${\displaystyle S=\left\{(x,p(z)/x^{n},z)\in \mathbb {C} ^{3},x\neq 0\right\}{\dot {\cup }}\left\{x=0,z=a_{1}\right\}{\dot {\cup }}\dots {\dot {\cup }}\left\{x=0,z=a_{m}\right\}}$

d. H. consists in principle of and copies of that are glued to it. ${\displaystyle S}$ ${\displaystyle \mathbb {C} ^{*}\times \mathbb {C} }$ ${\displaystyle m}$ ${\displaystyle \mathbb {C} }$

Automorphism group

The group of holomorphic automorphisms of a Danielewski surface, which has no singularities, behaves similarly to the known special case , that is, it is "large" in the sense that the elements that create the group cannot be explicitly specified. As in the case of , however, a dense subset of the automorphism group can be specifically described with the help of generalized shearings. ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \mathbb {C} ^{2}}$

Web links

• Automorphisms of Danielewski surfaces (pdf, engl.)