[Paper Review] New free divisors from old

Ragnar-Olaf Buchweitz, Aldo Conca|arXiv (Cornell University)|Nov 19, 2012

Algebraic Geometry and Number Theory3 references4 citations

TL;DR

This paper introduces new constructions and characterizations of free divisors in algebraic geometry, particularly focusing on weighted homogeneous and binomial free divisors. It presents a chain rule for generating quasihomogeneous free divisors, classifies triangular and binomial free divisors via Saito's criterion, and shows that homogeneous free divisors extend into the tangent bundle as free divisors, preserving linearity. The key contribution is a systematic method to generate families of free divisors from known ones, including via the discriminant matrix and Euler vector field annihilations.

ABSTRACT

We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and show that adding a normal crossing divisor to a smooth one will not succeed.

Motivation & Objective

• To develop systematic methods for constructing new free divisors from known ones, especially in the context of weighted homogeneous and quasihomogeneous polynomials.
• To classify binomial free divisors in terms of their algebraic structure and provide a complete characterization.
• To investigate whether any reduced polynomial can be a factor of a free divisor, and to show limitations of simple extension methods.
• To extend free divisors into the tangent bundle, preserving their freeness and linearity, using a generalized chain rule.
• To characterize free divisors annihilated by multiple Euler vector fields and relate this to the Buchsbaum-Rim complex.

Proposed method

• Use of the Saito matrix criterion: a free divisor $ f $ is characterized by the existence of an $ n \times n $ matrix $ A $ with $ \det(A) = f $ and $ (\nabla f)A \equiv 0 \mod (f) $.
• Application of the chain rule for free divisors: if $ f $ and $ g $ are free in disjoint variables, then $ fg(f+g) $ is also a free divisor.
• Construction of triangular free divisors by ensuring the Saito matrix has upper-triangular form, leading to explicit families like $ \prod_{j=2}^n (x_1^t + \cdots + x_j^t) $.
• Identification of binomial free divisors via a precise algebraic form: up to scaling and permutation, they must be of the form $ x_1\cdots x_n y^u z^t (y^\alpha \prod x_i^{a_i} + z^\beta \prod x_i^{b_i}) $ with $ \min(a_i,b_i) = 0 $, $ \alpha,\beta > 0 $, and $ u,t \in \{0,1\} $.
• Extension of a homogeneous free divisor $ f $ into the tangent bundle by forming $ f^* = \sum_i \frac{\partial f}{\partial x_i} y_i $, and showing $ f \cdot f^* $ is again a free divisor.
• Use of weighted Euler vector fields: replacing the standard Euler vector field with $ (w_1x_1, \dots, w_nx_n)^T $ in the discriminant matrix for weighted homogeneous cases.

Experimental results

Research questions

• RQ1 Can free divisors be systematically constructed from known ones using algebraic operations such as composition or extension?
• RQ2 What is the precise algebraic structure of a binomial free divisor in $ n+2 $ variables, and can it be fully characterized?
• RQ3 Under what conditions is a polynomial $ f $ annihilated by $ n-2 $ linearly independent Euler vector fields, and when does this imply freeness?
• RQ4 Can any reduced polynomial be embedded as a factor of a free divisor, and what obstructions exist to such extensions?
• RQ5 How do free divisors behave under extension into the tangent bundle, and does this preserve their freeness and linearity?

Key findings

• A polynomial $ f $ is a free divisor if it is annihilated by $ n-2 $ linearly independent Euler vector fields and the gradient $ \nabla f $ vanishes in the first homology of the Buchsbaum-Rim complex.
• The composition $ fg(f+g) $ is a free divisor whenever $ f $ and $ g $ are free divisors in disjoint sets of variables.
• The polynomial $ \prod_{j=2}^n (x_1^t + \cdots + x_j^t) $ is a free divisor for all $ t \geq 1 $, $ n \geq 2 $, due to its triangular Saito matrix.
• A binomial in $ n+2 $ variables is a free divisor if and only if it is, up to scaling and permutation, of the form $ x_1\cdots x_n y^u z^t (y^\alpha \prod x_i^{a_i} + z^\beta \prod x_i^{b_i}) $ with $ \min(a_i,b_i) = 0 $, $ \alpha,\beta > 0 $, and $ u,t \in \{0,1\} $.
• The product $ f \cdot f^* $, where $ f^* = \sum_i \frac{\partial f}{\partial x_i} y_i $, is a free divisor in $ 2n $ variables whenever $ f $ is a homogeneous free divisor, and this construction preserves linearity.
• For any weighted homogeneous free divisor $ f $ of degree $ d \neq 0 $, the product $ f \prod_{j=1}^m f^{*_{j}} $, where $ f^{*_{j}} = \sum_i y_{ij} \frac{\partial f}{\partial x_i} $, is a free divisor of weighted degree $ (m+1)d $, and remains linear if $ f $ is linear.