Lifting free divisors

Let $\varphi:X\to S$ 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 $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi$, then $f\circ \varphi=0$ 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 $\varphi:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi$ is the algebraic quotient $X\to S=X// G$ with $X// G$ smooth, we describe sufficient conditions for $T^1_{X/S}$ 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.

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