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On the smooth locus of aligned Hilbert schemes, the \(k\)-secant lemma and the general projection theorem. (English) Zbl 1262.14058

Let \(X \subset \mathbb{P}^N\) be a smooth complex quasiprojective variety. A line \(L \subset \mathbb{P}^N\) is said \(k\)-secant to \(X\) if the scheme \(L \cap X\) is finite of degree \(\geq k\). The locus of \(k\)-secant lines to \(X \subset \mathbb{P}^N\) is a classical object of study in projective geometry, naturally in relation with the study of the projections \(\pi:X \to X_1 \subset \mathbb{P}^{N-1}\) into \(\mathbb{P}^{N-1}\) from a general point of \(\mathbb{P}^N\). In particular, it is of interest to study the subscheme \(X_{\{k_1, \dots, k_r\}} \subset X_1\) formed by the points \(x \in X_1\) such that \(\pi^{-1}(x)\) contains \(r\) points \(\{x_1,\dots, x_r\}\) (possibly equal) with multiplicity greater than or equal to \(k_i\) in \(x_i\). In the paper under review it is proven (see General Projection Theorem, Theorem 1.1) that this scheme is pure dimensional of dimension \(N-1-\sum (k_i(N-n)-1)\). Moreover its singular locus is \(X_{\{k_1, \dots, k_r,1\}}\) and its normalization is smooth.
In order to be precise about the scheme structure of \(X_{\{k_1, \dots, k_r\}}\), the General Projection Theorem is presented as a consequence of a result stated in the language of Hilbert schemes of aligned points, naturally equipped with a map to the Grassmannian of lines \(G(1,N)\). This is Theorem 1.3, named Aligned Hilbert Scheme Theorem, where the schemes \(X_{\{k_1, \dots, k_r\}}\) can be related naturally with fibers of natural projections of incidence varieties constructed by means of the map from the Hilbert scheme to the Grassmannian (see Theorem 1.3 for details).
The Aligned Hilbert Scheme Theorem results to be a consequence of the Aligned ordered Hilbert Scheme Theorem where Hilbert schemes are substituted by ordered Hilbert schemes, parameterizing finite aligned subschemes supported in an ordered set of pints \((x_1,\dots, x_r)\) and with multiplicity \(k_i\) at \(x_i\). Being this ordered Hilbert scheme finite and flat over the non-ordered one, its smoothness implies smoothness of the non-ordered one.
An interesting section (Section 5) of examples, questions and conjectures (specially about the irreducibility of the locus of points of order \(k\) of a general projection) is also provided.

MSC:

14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry
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References:

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