凸集的并集与完整化来自单纯伪流形
Convex unions and completions from simplicial pseudomanifolds
AI总结：
研究通过单纯伪流形的泛化，探讨PL同胚对局部凸性的影响，揭示了收缩边对凸性空间的显著影响。
摘要：
尽管凸集的交集是凸的，但它们的并集却有相当复杂的性质。一些自然的出现场合包括涉及凸集边界的对偶论证和估值，这些具有类似欧拉特征的结构。然而，在某些情况下，凸性本身的重要性需要考虑。例如，这包括（保持或）正则性性质在toric簇上的除子在吹胀下的保持。在(限制的)共轭包络的情况下，这可以解释为在有理等价关系中存储的局部凸性数据之间的相互作用。我们考虑了对单纯伪流形的泛化，并用PL同胚的效果代替有理等价。将PL同胚分解为边细分和收缩，我们以局部凸性属性为条件，将合适的收缩点空间特征化为凸并集和完整化。这导致了某些外部边细分，使得起始边的“收缩空间”为空，这与预期的“增加的凸性”从边细分相悖。我们还获得了关于包含附近边的面的实现的强仿射/线性限制，这表明收缩某些附近的边会导致起始边的收缩空间非常大或非常小。至于边界行为，PL同胚对1-骨架上诱导的4-环的影响有平行关系。最后，我们发现PL同胚和悬垂对由线性参数系统存储的局部凸性属性的效应。这表明，单纯球体PL同胚于交叉多面体的边界在最自然的方式下存储了记录的局部凸性变化。
英文摘要：
While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler characteristic-like structure. However, there are certain settings where the convexity property itself is important to consider. For example, this includes (preservation of) positivity properties of divisors on toric varieties under blowdowns. In the case of (restrictions of) conormal bundles, this can be interpreted in terms of interactions between local convexity data stored in rational equivalence relations. We consider generalizations to realizations of simplicial pseudomanifolds and replace rational equivalence with effects of PL homeomorphisms. Decomposing the PL homeomorphisms into edge subdivisions and contractions, we characterize the space of suitable contraction points compatible with local convexity properties in terms of convex unions and completions. This gives rise to certain external edge subdivisions that make this ``contraction space'' of the starting edge empty, which is unexpected given the expected ``increased convexity'' from edge subdivisions. We also obtain strong affine/linear restrictions on realizations of facets containing nearby edges preserving local convexity. This implies that contracting certain nearby edges results in a very large or very small contraction space of the starting edge. As for boundary behavior, there are parallels between effects of PL homeomorphisms on induced 4-cycles in the 1-skeleton. Finally, we find effects of PL homeomorphisms and suspensions on analogues of local convexity properties stored by linear systems of parameters. This indicates that simplicial spheres PL homeomorphic to the boundary of a cross polytope store record local convexity changes in the most natural way.
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