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RESEARCH PRODUCT

Embeddings of Danielewski hypersurfaces

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In this thesis, we study a class of hypersurfaces in $\mathbb{C}^3$, called \emph{Danielewski hypersurfaces}. This means hypersurfaces $X_{Q,n}$ defined by an equation of the form $x^ny=Q(x,z)$ with $n\in\mathbb{N}_{\geq1}$ and $\deg_z(Q(x,z))\geq2$. We give their complete classification, up to isomorphism, and up to equivalence via an automorphism of $\mathbb{C}^3$. In order to do that, we introduce the notion of standard form and show that every Danielewski hypersurface is isomorphic (by an algorithmic procedure) to a Danielewski hypersurface in standard form. This terminology is relevant since every isomorphism between two standard forms can be extended to an automorphism of the ambiant space. (We show that this property does not hold for general Danielewski hypersurfaces.)Problems of stable equivalence and analytic equivalence are also studied. We construct examples of polynomials $P,Q\in\mathbb{C}[x,y,z]$ such that there does not exist an algebraic automorphism of $\mathbb{C}[x,y,z]$ which sends $P$ to $Q$, whereas these polynomials are equivalent via an automorphism of $\mathbb{C}[x,y,z,w]$.Most of these results are based on a precise picture of the sets of locally nilpotent derivations of the algebras of regular functions on the hypersurfaces $X_{Q,n}$, obtained using techniques developed by Makar-Limanov.