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Abstract The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least$$4$$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with$$1$$-liminal singularities (which are exactly the ordinary double points in dimension$$3$$but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneousk-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.
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Nearly magic rectangles
Abstract Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares ismagic rectangleswhich are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by oddnearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.
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- Award ID(s):
- 1935729
- PAR ID:
- 10116581
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Combinatorial Designs
- Volume:
- 27
- Issue:
- 9
- ISSN:
- 1063-8539
- Page Range / eLocation ID:
- p. 562-567
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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