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arXiv 1802.00622math.AG
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Hilbert点格式退化的几何学

The geometry of degenerations of Hilbert schemes of points

Martin G. Gulbrandsen, Lars H. Halle, Klaus Hulek, Ziyu Zhang

备注：
53 pages. To appear in J. Algebraic Geom
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摘要：

给定一个严格简单退化$f \colon X\to C$，前三位作者构造了$0$维子格式的相对阶$n$ Hilbert格式的退化$I^n_{X/C} \to C$.在本文中，我们调查这种退化的几何形状，特别是当纤维尺寸的$f$是最多$2$。在这种情况下，我们证明了$I^n_{X/C} \to C$是一个dlt模型。如果$f \colon X \to C$具有此属性，则这甚至是一个很好的最小dlt模型。我们计算的双重复杂的中央纤维$(I^n_{X/C})_0$，并与此有关的基本骨架的通用纤维。对于II型退化的$K3$表面，我们表明，堆栈${\mathcal I}^n_{X/C} \to C$进行无处退化的相对对数$2$形式。最后，我们讨论了我们的退化与永井结构的关系。

英文摘要：

Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of $K3$ surfaces we show that the stack ${\mathcal I}^n_{X/C} \to C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.

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