[Paper Review] Apolar Ideal and Normal Bundle of Rational Curves

Alessandro Bernardi|arXiv (Cornell University)|Mar 22, 2012

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper uses apolarity theory to characterize subvarieties in Hilbert schemes of rational curves in projective space, parameterized by the splitting type of their normal or restricted tangent bundle. By analyzing projections of the rational normal curve from linear subspaces, it establishes precise formulas for the splitting types of these bundles under geometric conditions on the center of projection, such as lying on secant spaces of specified dimensions.

ABSTRACT

As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves, defined by the splitting type of the normal bundle and the restricted tangent bundle.

Motivation & Objective

• To classify rational curves in P^m of degree n ≥ m by the splitting type of their normal bundle and restricted tangent bundle.
• To study subvarieties of the Hilbert scheme H^{m,n} parameterized by fixed splitting types of the normal and restricted tangent bundles.
• To use apolarity theory to relate geometric conditions on the center of projection (e.g., lying on secant spaces) to the splitting types of the normal and tangent bundles.
• To determine the codimension and irreducibility of such subvarieties in the Grassmannian of centers of projection.
• To provide explicit formulas for the splitting types of the normal and restricted tangent bundles under specific geometric assumptions on the projection center.

Proposed method

• Projecting the rational normal curve C_n ⊂ P^n from a linear subspace L ≅ P^{k-1} to obtain a rational curve π_L(C_n) ⊂ P^{n-k}.
• Using apolarity theory to analyze the syzygy module Syz(J(ν_n)) and the map (N^L_{n,k})^t, which encodes the normal bundle of the projected curve.
• Relating the rank and structure of the matrix N^L_{n,k} to the splitting type of the normal bundle via the condition rank(N^L_{n,k}) = 3k - r.
• Analyzing the kernel of N^L_{n,k} to detect when certain forms annihilate the curve, linking to the existence of secant spaces.
• Computing codimensions of loci of centers L lying on secant spaces (e.g., (k+1)-secant P^k) in the Grassmannian Gr(P^{k-1}, P^n) using incidence varieties.
• Applying results from Chiantini and Ciliberto on non-defectivity of secant varieties to compute codimensions and establish irreducibility of the loci.

Experimental results

Research questions

• RQ1 Under what geometric conditions on the center of projection L ⊂ P^n does the normal bundle of the projected rational curve π_L(C_n) ⊂ P^{n-k} have a specific splitting type?
• RQ2 How does the presence of L in a (k+1)-secant or (k+2)-secant space to the rational normal curve affect the splitting type of the normal bundle?
• RQ3 What is the codimension in the Grassmannian Gr(P^{k-1}, P^n) of the locus of centers L such that the restricted tangent bundle T P^{n-k}|_{π_L(C_n)} has a given splitting type?
• RQ4 Can apolarity theory be used to characterize the splitting type of the normal bundle via algebraic conditions on the annihilating forms of points in L?
• RQ5 When is the subvariety of the Hilbert scheme parameterizing curves with a fixed splitting type of the normal or restricted tangent bundle irreducible and of a given codimension?

Key findings

• If L ⊂ P^{k+1} is a (k+2)-secant to C_n ⊂ P^n and n−1 ≤ 3k ≤ 3(n−3), then the normal bundle of the projected curve π_L(C_n) ⊂ P^{n-k} splits as O(n+2)^{n−k−2} ⊕ O(n+1+2k).
• If L ⊂ P^k is a (k+1)-secant to C_n and 2k < n−1, then the restricted tangent bundle T P^{n-k}|_{π_L(C_n)} splits as O(n+1)^{n−k−1} ⊕ O(n+1+k).
• The variety of centers L ⊂ P^n lying on some (k+1)-secant P^k to C_n has codimension kn − k² − k − 1 in Gr(P^{k−1}, P^n), and this locus supports an irreducible subvariety of the Hilbert scheme parameterizing curves with the given tangent bundle splitting type.
• For L contained in a (k+1)-secant P^k, the rank of the matrix N^L_{n,k} satisfies rank(N^L_{n,k}) = 3k − r with 1 < r ≤ 2k−1 if and only if the normal bundle is O(n+3)^{2k−2r} ⊕ F' with deg(F'^∨(n+2)) = −2k.
• When rank(N^L_{n,k}) = k+1, the normal bundle splits as O(n+2+2k), which corresponds to the case where L lies on a (k+2)-secant P^{k+1}.
• The codimension of the locus of centers L lying on some (n−2)-secant P^{n−3} to C_n is 3k − n + 1, which is negative when 3k < n−1, implying the condition is generic in that range.