[Paper Review] Chern-Schwartz-MacPherson Class of Determinantal Varieties

Xiping Zhang|arXiv (Cornell University)|May 17, 2016

Algebraic structures and combinatorial models3 citations

TL;DR

This paper computes the Chern-Schwartz-MacPherson (CSM) class of determinantal varieties $\alpha_{m,n,k}$, defined as the set of $m \times n$ matrices with kernel dimension at least $k$, using a resolution of singularities realized as a projective bundle over a Grassmannian. The key contribution is an explicit formula for $c_{\text{SM}}(\alpha_{m,n,k})$ that leverages functoriality and Grassmannian geometry, implementable in Macaulay2, and includes conjectures on effectivity and vanishing in the nonsingular part of the loci.

ABSTRACT

For $m\geq n$, let $ au_{m,n,k}$ be the set of $m imes n$ matrices with kernel dimension $\ge k$. It is viewed as a subvariety of the projective space $\mathbb{P}^{mn-1}$ of nonzero $m imes n$ matrices up to scalar. In most cases $ au_{m,n,k}$ is singular, and its singular locus is $ au_{m,n,k+1}$. In this note we compute its Chern-Schwartz-MacPherson class $c_{SM}( au_{m,n,k})$ by using a resolution of singularities $ u\colon \hat au_{m,n,k} o au_{m,n,k} $, which may also be realized as a projective bundle over a Grassmannian. By the functoriality of $c_{SM}$ class, we can relate $c_{SM}( au_{m,n,k})$ to the pushforward $ u_*c_{SM}(\hat au_{m,n,k})$, and the Grassmanian plays an important role in the computation. Our formula can be easily implemented in Macaulay2. On the basis of explicit computations in low dimension, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the $c_{SM}$ classes of the nonsingular part of the loci $ au_{m,n,k}$.

Motivation & Objective

• To compute the Chern-Schwartz-MacPherson class of the singular determinantal variety $\alpha_{m,n,k}$, the locus of $m \times n$ matrices with kernel dimension at least $k$.
• To resolve the singularities of $\alpha_{m,n,k}$ using a geometric construction that realizes the resolution as a projective bundle over a Grassmannian.
• To exploit the functoriality of the CSM class to relate the class of the singular variety to the pushforward of the CSM class of its resolution.
• To derive a formula for $c_{\text{SM}}(\alpha_{m,n,k})$ that is computationally feasible, particularly implementable in Macaulay2.
• To formulate conjectures on the effectivity of the CSM classes and the vanishing of specific terms in the nonsingular part of the loci $\alpha_{m,n,k}$.

Proposed method

• Use a resolution of singularities $\tilde{\alpha}_{m,n,k} \to \alpha_{m,n,k}$, which is constructed as a projective bundle over a Grassmannian parametrizing $k$-dimensional subspaces.
• Apply the functoriality property of the Chern-Schwartz-MacPherson class to express $c_{\text{SM}}(\alpha_{m,n,k})$ as the pushforward of $c_{\text{SM}}(\tilde{\alpha}_{m,n,k})$ under the resolution map.
• Leverage the known intersection theory on Grassmannians to compute the pushforward via the projective bundle structure.
• Utilize the structure of the resolution to reduce the computation of the CSM class to a problem in Schubert calculus on the Grassmannian.
• Implement the resulting formula in Macaulay2 for explicit computation in low-dimensional cases.
• Base conjectures on observed patterns in low-dimensional computations regarding the effectivity and vanishing of terms in the CSM class of the nonsingular part of $\alpha_{m,n,k}$.

Experimental results

Research questions

• RQ1 How can the Chern-Schwartz-MacPherson class of the singular determinantal variety $\alpha_{m,n,k}$ be computed despite its singularities?
• RQ2 What is the role of the Grassmannian in resolving the singularities of $\alpha_{m,n,k}$ and enabling the computation of its CSM class?
• RQ3 Can the CSM class of $\alpha_{m,n,k}$ be expressed in terms of pushforwards from a resolution that is a projective bundle over a Grassmannian?
• RQ4 What structural properties, such as effectivity and vanishing, do the CSM classes of the nonsingular parts of $\alpha_{m,n,k}$ exhibit?
• RQ5 How can the resulting formula be systematically implemented in computational algebraic geometry software like Macaulay2?

Key findings

• The Chern-Schwartz-MacPherson class $c_{\text{SM}}(\alpha_{m,n,k})$ is computed via a resolution that is a projective bundle over a Grassmannian, enabling explicit calculation.
• The formula for $c_{\text{SM}}(\alpha_{m,n,k})$ is derived using the functoriality of the CSM class and pushforward along the resolution map.
• The resolution map $\tilde{\alpha}_{m,n,k} \to \alpha_{m,n,k}$ allows the CSM class of the singular variety to be expressed as the pushforward of the CSM class of the smooth resolution.
• The resulting formula is implementable in Macaulay2, facilitating explicit computations in low-dimensional cases.
• Based on low-dimensional computations, the authors formulate conjectures on the effectivity of the CSM classes and the vanishing of specific terms in the nonsingular part of $\alpha_{m,n,k}$.
• The Grassmannian plays a central role in organizing the geometry of the resolution and enabling the computation through intersection-theoretic techniques.