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An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
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Topology change induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - 4)^2 + z …
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Diffeomorphism induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{align}
Obvio …