Lifting free divisors

Abstract

Let $\phi :X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T_{X/S}^1$ of $\phi$ is a Cohen-Macaulay ${\mathcal{O}}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\phi$, then $f\circ \phi =0$ defines a free divisor on $X$; this is generalized in several directions. Among applications we recover a result of Mond-van Straten, generalize a construction of Buchweitz-Conca, and show that a map $\phi :{\mathbb{C}}^{n+1}\to {\mathbb{C}}^n$ with critical set of codimension $2$ has a $T_{X/S}^1$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\phi$ is the algebraic quotient $X\to S=X//G$ with $X//G$ smooth, we describe sufficient conditions for $T_{X/S}^1$ to be Cohen-Macaulay of codimension $2$. In one such case, a free divisor on ${\mathbb{C}}^{n+1}$ lifts under the operation of "castling" to a free divisor on ${\mathbb{C}}^{n(n+1)}$, partially generalizing work of Granger-Mond-Schulze on linear free divisors. We give several other examples of such representations.

30 pages. Many minor changes from v1 in response to a thorough review process. To appear in Proc. London Math. Soc. This version differs from the final published version

References in corpus (1)

• On the formal structure of logarithmic vector fields

Cited by in corpus (1)

• Free and nearly free surfaces in $P^3$