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Whitney equisingularity in family of generically reduced curves Da Silva, Otoniel Nogueira


We say that two germs of analytic sets $(M,0)$ and $(N,0)$ in $(\mathbb{C}^n,0)$ have the same embedded topological type or are topologically equivalent if there exists a germ of homeomorphism $\varphi : (\mathbb{C}^n,M,0) \rightarrow (\mathbb{C}^n,N,0)$. We consider a topologically trivial (flat) family of generically reduced curves $p : (X,0) \rightarrow (\mathbb{C},0)$ in $(\mathbb{C}^n \times \mathbb{C},0)$ with a section $\sigma: (\mathbb{C},0)\rightarrow (X,0)$ and fibers $(X_t,\sigma(t)):=p^{-1}(t)$. In this case, we know that the special curve $(X_0,\sigma(0))$ and the generic curve $(X_t,\sigma(t))$ are topologically equivalent, for all $t$. So we can ask in what conditions the tangent cone $C_{(X_t,\sigma(t))}$ of $(X_t,\sigma(t))$ does not change the topological type under topological trivial deformations of $(X_0,\sigma(0))$. In others words, we study the following question: if $p:(X,0)\rightarrow (\mathbb{C},0)$ is a topological trivial family of generically reduced curves, under what conditions the Zariski tangent cones $C_{(X_0,\sigma(0))}$ and $C_{(X_t,\sigma(t))}$ are homeomorphic (Joint work with J. Snoussi and A. Giles Flores)

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