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Search: MSC category 13F55 ( Stanley-Reisner face rings; simplicial complexes [See also 55U10] )

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1. CJM Online first

Choi, Suyoung; Park, Hanchul
Wedge operations and torus symmetries II
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. We refer to $K$ as a seed if $K$ cannot be obtained by wedgings. First, we show that, for a fixed positive integer $\ell$, there are at most finitely many seeds of Picard number $\ell$ supporting characteristic maps. As a corollary, the conjecture proposed by V.V. Batyrev in 1991 is solved affirmatively. Second, we investigate a systematic method to find all characteristic maps on $K(J)$ using combinatorial objects called (realizable) puzzles that only depend on a seed $K$. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Keywords:puzzle, toric variety, simplicial wedge, characteristic map
Categories:57S25, 14M25, 52B11, 13F55, 18A10

2. CJM 2009 (vol 61 pp. 888)

Novik, Isabella; Swartz, Ed
Face Ring Multiplicity via CM-Connectivity Sequences
The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring $\field[\Delta]$ via the Cohen--Macaulay connectivity of the skeletons of $\Delta$.

Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35

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