Embeddings of hypersurfaces in affine spaces

Details Embeddings of hypersurfaces in affine spaces Embeddings of hypersurfaces in affine spaces

2000-10-22

arxiv.org/abs/math/0010211v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

In this paper, we address the following two general problems: given two algebraic varieties in ${\bf C}^n$, find out whether or not they are (1) isomorphic; (2) equivalent under an automorphism of ${\bf C}^n$. Although a complete solution of either of these problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one, and use it to construct numerous examples of isomorphic, but inequivalent algebraic varieties in ${\bf C}^n$. Among other things, we establish the following interesting fact: for isomorphic hypersurfaces $\{p(x_1,...,x_n)=0\}$ and $\{q(x_1,...,x_n)=0\}$, the number of zeros of $grad(p)$ might be different from that of $grad(q)$.

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