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On the geometry of polar varieties. (English) Zbl 1186.14060

Let \(F_1, \dots, F_p \in \mathbb{Q}[x_1, \dots, x_n]\), \(S=V(F_1, \dots, F_p)\subseteq {\mathbb C}^n\) and \(S_{\mathbb R}= S\cap {\mathbb R}^n\). Let \(J(F_1, \dots, F_p)=\left(\frac{\partial F_i}{\partial x_j}\right)\) be the Jacobian matrix. Let \(\leq i\leq n-p\) and \(\alpha=(a_{kl})_{1\leq k\leq n-p-i-l}\) be a complex \(n-p-i+1 \times n+1\)-matrix and suppose that \(\alpha_\ast=(a_{kl})_{1\leq k\leq n-p-i+1}\) has maximal rank.
In the case \((a_{1,0},\dots, a_{n-p-i+1,0})=0\) resp. \(\neq 0\;\underline{K}(\alpha)\) (resp. \(\overline{K}(\alpha))\) denotes the \(n-p-i\)-dimensional linear subvarieties of \({\mathbb P}^n\) which for \(1\leq k\leq n-p-i+1\) are spanned by the points \((a_{k,0}:a_{k,1}:\dots :a_{k,n})\).
The classic and the dual \(i\)-th polar varieties of \(S\) associated with \(\underline{K}(\alpha)\) and \(\overline{K}(\alpha)\) are defined as the closures of the loci of the regular points of \(S\) where all \((n-i+1)\)-minors of the matrix \[ \left[\begin{matrix} & J(F_1, \dots, F_p) & \\ a_{1,1} & & a_{1,n}\\ \vdots & & \vdots\\ a_{n-p-i+1, 1} & \cdots & a_{n-p-i+1, n} \end{matrix} \right] \] resp.
\[ \left[\begin{matrix} & J(F_1, \dots, F_p) & \\ a_{1,1}-a_{1,0} x_1 & & a_ {1,n}-a_{1,0}x_n\\ \vdots & & \vdots \\ a_{n-p-i+1, 1}-a_{n-p-i+1,0} x_1 &\cdots & a_{n-p-i+1, n} - a_{n-p-i+1,0} x_n \end{matrix} \right] \] vanish. These polar varieties are denoted by \(W_{\underline{K}(\alpha)}(S)\) and \(W_{\overline{K}(\alpha)}(S_{\mathbb R})=W_{\overline{K}(\alpha)}(S)\cap {\mathbb R}^n\). The main result of the first section is the following:
Let \(1\leq i\leq n-p\) and \(C\) be a connected component of \(S_{\mathbb R}\) containing a regular point. Then there exists a non-empty open (with respect to the Euclidean topology) subset \({\mathcal O}_C^{(i)}\subseteq {\mathbb R}^{(n-p-i+1)\times n}\) such that any matrix \(\alpha\in{\mathcal O}_C^{(i)}\) has maximal rank and such that the real dual polar variety \(W_{\overline{K}(\alpha)}(S_{\mathbb R})\) is generic and contains a regular point of \(C\).
In the next section it is proved that in case \(S\) is smooth the generic classic and dual polar varieties are normal. Hence the generic polar varieties of \(S\) are both, normal and Cohen–Macaulay. It is shown that generic polar varieties may become singular at smooth points of \(S\). The new concept of meagerly generic polar varieties is introduced and a degree estimate for them in terms of the degrees of generic polar varieties is given. The statements are illustrated by examples and a computer experiment.

MSC:

14P05 Real algebraic sets
14B05 Singularities in algebraic geometry
14Q10 Computational aspects of algebraic surfaces
14Q15 Computational aspects of higher-dimensional varieties
68W30 Symbolic computation and algebraic computation

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