A genus one curve defined over $\mathbb{Q}$ which has points over ${\mathbb{Q}}_p$ for all primes $p$ may not have a rational point. It is natural to study the classes of $\mathbb{Q}$-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 $\mathbb{Q}$

We study the irregularity sheaves attached to the $A$-hypergeometric D-module $M_A(\beta )$ introduced by I. M. Gel'fand and others in [GGZ], [GZK], where $A\in {\mathbb{Z}}^{d\times n}$ is pointed of full rank and $\beta \in {\mathbb{C}}^d$. 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 $L$ on torus-equivariant generators. To this end, we introduce the $(A,L)$-umbrella, a cell complex determined by $A$ and $L$, and identify its facets with the components of the associated graded ring.

We then establish a correspondence between the full $(A,L)$-umbrella and the components of the $L$-characteristic variety of $M_A(\beta )$. We compute in combinatorial terms the multiplicities of these components in the $L$-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities.

We deduce from this that slopes of $M_A(\beta )$ are combinatorial, independent of $\beta$, and in one-to-one correspondence with jumps of the $(A,L)$-umbrella. This confirms a conjecture of B. Sturmfels and gives a converse of a theorem of R. Hotta [Ho, Chap. II, §6.2, Th.]: $M_A(\beta )$ is regular if and only if $A$ defines a projective variety

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to $\mathfrak{sl}(2)$ 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])