论文标题
E $ _7 $ -MODEL
Flops and Fibral Geometry of E$_7$-models
论文作者
论文摘要
E $ _7 $ -WEIERSTRASS模型的猜想具有八个不同的毛茸茸的分辨率，其插图图是E $ _8 $的Dynkin图。在以前的工作中，我们明确构建了四个不同的分辨率，为此，flop图构成了d $ _4 $ $ diagram。 本文的目的是探索已解决的E $ _7 $模型的属性，这些属性并非不变。特别是，我们检查了纤维退化，确定纤维分裂至同构，并研究侵犯在基部的某些Codimension-Three基因座上出现的平稳性，在该基础上，纤维的成分从理性曲线生长到理性表面。对于每种捕食者的分辨率，我们计算由第二个Chern类诱导的三重相交多项式和线性形式，以及全态和普通的Euler特征，以及每个外部分裂的签名。我们确定了破坏纤维固定度的理性表面的同构类别。此外，我们明确地表明，d $ _4 $ flops对应于$ \ mathbb {c}^3 $ sodient the klein four four tour the klein four tour the Orbifold的毛茸茸的分辨率。
An E$_7$-Weierstrass model is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E$_8$. In previous work, we explicitly constructed four distinct resolutions, for which the flop diagram formed a D$_4$ sub-diagram. The goal of this paper is to explore those properties of a resolved E$_7$-model which are not invariant under flops. In particular, we examine the fiber degenerations, identify the fibral divisors up to isomorphism, and study violation of flatness appearing over certain codimension-three loci in the base, where a component of the fiber grows in dimension from a rational curve to a rational surface. For each crepant resolution, we compute the triple intersection polynomial and the linear form induced by the second Chern class, as well as the holomorphic and ordinary Euler characteristics, and the signature of each fibral divisor. We identify the isomorphism classes of the rational surfaces that break the flatness of the fibration. Moreover, we explicitly show that the D$_4$ flops correspond to the crepant resolutions of the orbifold given by $\mathbb{C}^3$ quotiented by the Klein four-group.
