[Paper Review] The k-secant lemma and the general projection theorem

Laurent Gruson, Christian Peskine|arXiv (Cornell University)|Oct 12, 2010

Algebraic Geometry and Number Theory3 references1 citations

TL;DR

This paper investigates the singular locus of the Hilbert scheme parametrizing aligned, finite, degree-∑k_i subschemes on a smooth quasi-projective variety X ⊂ ℝ_N. By analyzing the expected dimension 2N−2+r−(∑k_i)(N−n), it proves that the lines through points where the Hilbert scheme fails to be smooth of expected dimension do not fill ℝ_N, establishing a key geometric constraint on the singular locus.

ABSTRACT

Let X be a smooth, connected, dimension n, quasi-projective variety imbedded in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with multiplicities k_i at points x_i (possibly coinciding). The expected dimension of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that the lines carrying this locus do not fill up \PP_N

Motivation & Objective

• To understand the structure of the Hilbert scheme H_{k_1,...,k_r}(X) parametrizing finite, aligned subschemes of X with specified multiplicities.
• To analyze the locus where this Hilbert scheme fails to be smooth of the expected dimension.
• To determine whether the lines passing through such singular points fill the ambient projective space ℝ_N.
• To establish geometric constraints on the singular locus of the Hilbert scheme via projection and dimension-theoretic arguments.

Proposed method

• The paper considers the Hilbert scheme H_{k_1,...,k_r}(X) of finite, degree-∑k_i subschemes on a smooth, connected, n-dimensional quasi-projective variety X embedded in ℝ_N.
• It computes the expected dimension of H_{k_1,...,k_r}(X) as 2N−2+r−(∑k_i)(N−n), based on deformation-theoretic expectations.
• It studies the singular locus where the Hilbert scheme fails to be smooth of this expected dimension.
• Using geometric and dimension-theoretic arguments, it examines the lines passing through points in this singular locus.
• It proves that these lines do not fill ℝ_N, implying the singular locus is contained in a proper subvariety of ℝ_N.
• The argument relies on the interplay between the geometry of X, the Hilbert scheme, and the behavior of secant lines in projective space.

Experimental results

Research questions

• RQ1 Does the singular locus of H_{k_1,...,k_r}(X) span the ambient projective space ℝ_N?
• RQ2 What is the dimension of the union of lines passing through points where H_{k_1,...,k_r}(X) is not smooth of expected dimension?
• RQ3 How does the expected dimension formula 2N−2+r−(∑k_i)(N−n) constrain the geometry of the Hilbert scheme?
• RQ4 Can the Hilbert scheme fail to be smooth at points whose secant lines cover ℝ_N?
• RQ5 What geometric obstructions prevent the singular locus from being dense in ℝ_N?

Key findings

• The Hilbert scheme H_{k_1,...,k_r}(X) has an expected dimension of 2N−2+r−(∑k_i)(N−n).
• The locus of points where H_{k_1,...,k_r}(X) fails to be smooth of expected dimension is not dense in ℝ_N.
• The lines passing through points in this singular locus do not fill the ambient projective space ℝ_N.
• This implies that the singular locus is contained in a proper subvariety of ℝ_N, imposing a strong geometric constraint.
• The result establishes a general projection theorem by showing that the singular locus cannot dominate ℝ_N via secant lines.
• The k-secant lemma is used to control the dimension of secant lines through singular points, leading to the non-filling conclusion.