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A genus one curve defined over which has points over for all primes may not have a rational point. It is natural to study the classes of -extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of
We study the irregularity sheaves attached to the -hypergeometric D-module introduced by I. M. Gel'fand and others in [GGZ], [GZK], where is pointed of full rank and . More precisely, we investigate the slopes of this module along coordinate subspaces.
In the process, we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector on torus-equivariant generators. To this end, we introduce the -umbrella, a cell complex determined by and , and identify its facets with the components of the associated graded ring.
We then establish a correspondence between the full -umbrella and the components of the -characteristic variety of . We compute in combinatorial terms the multiplicities of these components in the -characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities.
We deduce from this that slopes of are combinatorial, independent of , and in one-to-one correspondence with jumps of the -umbrella. This confirms a conjecture of B. Sturmfels and gives a converse of a theorem of R. Hotta [Ho, Chap. II, §6.2, Th.]: is regular if and only if defines a projective variety
Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to and its standard representation. Our construction is related to that of Seidel and Smith [SS] by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology (see [Kh1])
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