A Lefschetz hyperplane theorem for Mori dream spaces
A Lefschetz hyperplane theorem for Mori dream spaces
复制标题
森梦空间的 Lefschetz 超平面定理
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发表时间：
2008
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通讯作者：
Shin
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作者：
Shin
设X是一个≥4维的光滑Mori梦空间，我们证明了，如果X满足一个适当的Git条件，我们称之为小不稳定轨迹，则X的每个光滑充分因子Y也是Mori梦空间。此外，限制映射标识了X和Y的Néron-Severi空间，在这个标识下，Y的每个Mori腔都是X的一些Mori腔的并，Y的NEF锥与X的NEF锥相同。这个Lefschetz型定理使人们能够通过取周围Mori梦空间的“Mori梦超曲面”来构造许多Mori梦空间的例子，只要它满足Git条件。为了证明这一点，我们证明了Git条件在取乘积和取至少三个线丛的直和的投影丛下是稳定的，并且在X是环面的情况下，我们证明了该条件等价于X的扇形是2-邻域的。
Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron–Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking “Mori dream hypersurfaces” of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.