We examine the extremal rays of the cone of dominant weights $(μ, \widehatμ)$ for groups $G\subseteq \widehat G$ for which there exists $N \gg0$ such that $$ \left(V(Nμ)\otimes V(N\widehat μ)\right)^G\ne (0). $$ We exhibit formulas for a class of rays ("type I") on any regular face of the cone. These rays are identified thanks to a generalization of Fulton's conjecture, which we prove along the way. We verify that the remaining rays ("type II") on the face are the images of extremal rays for a smaller cone under a certain map, whose formula is given. A procedure is given for finding the rays of the cone not on any regular face. This is a generalization of the work of Belkale and Kiers on extremal rays for the saturated tensor cone; the specialization is given by $\widehat G = G\times G$ with the diagonal embedding of $G$. We include several examples to illustrate the formulas.
Diagram Method for 3-Folds and its Application to Kahler Cone and Picard Number of Calabi-Yau 3-Folds, I. with Appendix by Vyacheslav V. Shokurov: "Anticanonical boundedness for curves"
我们证明了一般的图解法定理对具有Q-阶乘奇点的相当大的一类3-重有效（见基本定理1.3.2和定理2.2.6）。这给出了我们关于具有Q-阶乘终端奇点的Fano 3-folds的结果（预印本alg-geom/9311007）的推广。作为应用，我们得到了关于三维Calabi-Yau流形X的如下结果：假设Picard数ρ（X）> 93.则以下两种情况之一成立：（i）X上存在一个小的极值射线;（ii）存在一个nef元素h使得h^3=0（因此，nef锥NEF（X）和三次相交超曲面W_X有一个公共点;这里，我们不主张h是有理的！）。作为推论，我们得到：设X是三维Calabi-Yau流形。假设nef锥NEF（X）是有限多面体，并且X没有小的极值射线。若ρ（X）>93，则存在有理nef元h，且h^3=0。为了证明关于Calabi-Yau流形的这些结果，我们还使用了V. V. Shokurov关于除极射线长度的一个结果（见附录）。因此，我们应该把上面关于Calabi-Yau 3-folds的结果看作是我们与V. V. Shokurov的共同结果。我们还讨论了上述结果的推广到所谓的Q-factorial模型的Calabi-Yau 3-folds，有时允许涉及非多面体的情况下，小的极值射线的游戏。
