arXiv 1310.78732013-10math.AG

提升自由因子

Lifting free divisors

Ragnar-Olaf Buchweitz, Brian Pike

备注：: 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
DOI：: 10.1112/plms/pdw013

摘要：

设$φ:X\to S$是光滑复解析空间之间的一个态射，$f=0$定义了$S$上的一个自由因子.我们证明了，如果变形空间$T^1_{X/S}$的$φ$是一个科恩-麦考利$\mathcal{O}_X$ -模的余维2，和所有的对数向量场$f=0$电梯通过$φ$，然后$f\circ φ=0$定义一个自由因子$X$ ;这是推广到几个方向。在应用中，我们恢复了Mond-van Straten的一个结果，推广了Buchweitz-Conca的一个构造，并证明了具有余维$2$临界集的映射$φ:\mathbb{C}^{n+1}\to \mathbb{C}^n$有一个具有所需性质的$T^1_{X/S}$.最后，如果$X$是约化复代数群$G$的表示，$φ$是$X// G$光滑的代数商$X\to S=X// G$，我们给出了$T^1_{X/S}$是余维$2$的Cohen-Macaulay的充分条件.在一个这样的情况下，$\mathbb{C}^{n+1}$上的自由因子在“易位”操作下提升为$\mathbb{C}^{n(n+1)}$上的自由因子，部分推广了Granger-Mond-Schulze关于线性自由因子的工作。我们给出了其他几个这样的代表性的例子。

英文摘要：

Let $φ:X\to S$ 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 $φ$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $φ$, then $f\circ φ=0$ 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 $φ:\mathbb{C}^{n+1}\to \mathbb{C}^n$ 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 $φ$ 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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英文摘要：

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